ARNOLD PIZER rochester problib from CVS Summer 2003doyle/docs/cs.old/raw/ww/pdf-summaries… · ARNOLD PIZER rochester problib from CVS Summer ... determine whether it has a unique

ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra1Systems due 1/1/09 at 2:00 AM

1.(1 pt) setLinearAlgebra1Systems/ur la 1 3.pgSolve the system using substitution�

x � 5y � 52x � 9y � 1

x �y �

2.(1 pt) setLinearAlgebra1Systems/ur la 1 4.pgSolve the system using substitution� � 6x � 5y �� 63� 3x � 3y �� 36x �y �

3.(1 pt) setLinearAlgebra1Systems/ur la 1 4a.pgSolve the system using elimination� � 6x � 3y � 21

5x � 5y � 5x �y �

4.(1 pt) setLinearAlgebra1Systems/ur la 1 5.pgSolve the system using elimination�� 4x � 3y � 3z � 13

1x � 2y � 4z �� 71x � 1y � 4z � 8

x �y �z �

5.(1 pt) setLinearAlgebra1Systems/ur la 1 23.pgWrite the system �� 11y � 4z �� 5� 4x � 2y � 9� 3x � 10y � 3z �� 6

in matrix form. �� xyz

�� 6.(1 pt) setLinearAlgebra1Systems/ur la 1 7.pg

Write the augmented matrix of the system�� 84y � 10z �� 7� 86x � 15z �� 507x � 6y � z � 33 ��

7.(1 pt) setLinearAlgebra1Systems/ur la 1 1.pgPerform one step of row reduction, in order to calculate the val-ues for x and y by back substitution. Then calculate the values

for x and for y. Also calculate the determinant of the originalmatrix.

You can let webwork do much of the calculation for you ifyou want (e.g. enter 45-(56/76)(-3) instead of calculating thevalue out). You can also use the preview feature in order tomake sure that you have used the correct syntax in entering theanswer.

[Note– since the determinant is unchanged by row reductionit will be easier to calculate the determinant of the row reducedmatrix.]� � 14 32

16 2 � �xy � � �

52 �� 14 32

0 � �xy � � � � 3 �

x �y �det �

8.(1 pt) setLinearAlgebra1Systems/ur la 1 2.pgPerform one step of row reduction, in order to calculate the val-ues for x and y by back substitution. Then calculate the valuesfor x and for y. Also calculate the determinant of the originalmatrix.

You can let webwork do much of the calculation for you ifyou want (e.g. enter 45-(56/76)(-3) instead of calculating thevalue out). You can also use the preview feature in order tomake sure that you have used the correct syntax in entering theanswer.

This problem has rather difficult complex calculations.[Note– since the determinant is unchanged by row reduction

it will be easier to calculate the determinant of the row reducedmatrix.]� � 3 � i 5 � 4i� 1 � 2i 5 � 2i � �

xy � � �

16 � 52i41 � 8i �� 3 � i 5 � 4i

0 � �xy � � �

16 � 52i �x �y �det �

9.(1 pt) setLinearAlgebra1Systems/ur la 1 4b.pgSolve the system using matrices (row operations)� � 2x � 5y � 28

1x � 9y �� 53x �y �

10.(1 pt) setLinearAlgebra1Systems/ur la 1 5a.pgSolve the system using matrices (row operations)�� 6x � 3y � 5z �� 34� 3x � 2y � 6z �� 15� 6x � 3y � 2z � 37x �

1

y �z �

11.(1 pt) setLinearAlgebra1Systems/ur la 1 16.pgSolve the system �� x � y �� 1

5x � 3y � 1114x � 10y � 34

x �y �

12.(1 pt) setLinearAlgebra1Systems/ur la 1 17.pgSolve the system�� x1 � 4x3 � 2x4 �� 33

x2 � 4x3 � 4x4 � 272x1 � 3x2 � 22x3 � 16x4 �� 161� x2 � 4x3 � 9x4 �� 27

x1 �x2 �x3 �x4 �

13.(1 pt) setLinearAlgebra1Systems/ur la 1 6.pg

For each system, determine whether it has a unique solution(in this case, find the solution), infinitely many solutions, or nosolutions.

1. � � 1x � 4y � 05x � 3y � 0� A. Unique solution: x �� 5 � y � 8� B. Infinitely many solutions� C. Unique solution: x �� 3 � y �� 1� D. Unique solution: x � 0 � y � 0� E. No solutions� F. None of the above

2. �3x � 4y � 17� 9x � 12y �� 50� A. Infinitely many solutions� B. Unique solution: x � 0 � y � 0� C. Unique solution: x �� 50 � y � 17� D. Unique solution: x � 17 � y �� 50� E. No solutions� F. None of the above

3. � � 1x � 3y � 6� 3x � 9y � 18� A. Infinitely many solutions� B. No solutions� C. Unique solution: x � 0 � y � 0� D. Unique solution: x �� 6 � y � 0� E. Unique solution: x � 6 � y � 18� F. None of the above

4. �5x � 6y � 43x � 3y � 9� A. Unique solution: x � 0 � y � 0� B. Unique solution: x �� 1 � y � 2� C. Unique solution: x � 2 � y �� 1� D. Infinitely many solutions� E. No solutions� F. None of the above

14.(1 pt) setLinearAlgebra1Systems/ur la 1 22.pgThe reduced row-echelon forms of the augmented matrices offour systems are given below. How many solutions does eachsystem have?

1.

�1 0 0 40 0 1 � 8 �� A. Unique solution� B. Infinitely many solutions� C. No solutions� D. None of the above

2.

1 0 120 1 90 0 0

�� A. Unique solution� B. Infinitely many solutions� C. No solutions� D. None of the above

3.

1 0 18 00 1 3 00 0 0 1

�� A. Unique solution� B. No solutions� C. Infinitely many solutions� D. None of the above

4.

�� 1 0 � 7 00 1 0 00 0 0 10 0 0 0

�� A. Infinitely many solutions� B. Unique solution� C. No solutions� D. None of the above

15.(1 pt) setLinearAlgebra1Systems/ur la 1 4c.pgSolve the system by using Cramer’s Rule.�

4x � 7y �� 545x � 6y �� 51

x �y �

16.(1 pt) setLinearAlgebra1Systems/ur la 1 8.pgDetermine the value of h such that the matrix is the augmentedmatrix of a consistent linear system.

2

�4 � 7� 8 14 �� h

5 �h �

17.(1 pt) setLinearAlgebra1Systems/ur la 1 9.pgDetermine the value of h such that the matrix is the augmentedmatrix of a linear system with infinitely many solutions.�

8 � 424 h �� 5

15 �h �

18.(1 pt) setLinearAlgebra1Systems/ur la 1 11.pgSolve the system

�2x � 1y � a1x � 1y � b

x �y �

19.(1 pt) setLinearAlgebra1Systems/ur la 1 12.pgDetermine the value of k for which the system�� x � y � 3z � 2

x � 2y � 3z � 35x � 12y � kz � 18

has no solutions.k �

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�

UR

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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra2SystemsApplications due 1/2/09 at 2:00 AM

1.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 1.pgSteve and Tonya are brother and sister. Steve has five times asmany sisters as brothers, and Tonya has twice as many sisters asbrothers. How many boys and girls are there in this family?

Answer: boys and girls.

2.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 2.pgFind the quadratic polynomial whose graph goes through thepoints � � 1 � 3 !"�#� 0 � 3 !"� and � 1 � 7 ! .f � x !$� x2 � x �

3.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 3.pgFind the polynomial of degree 4 whose graph goes through thepoints � � 2 �%� 38 !"�&� � 1 � 5 !"�#� 0 � 10 !"�#� 1 � 19 !"� and � 2 � 26 !"'f � x !$� x4 � x3 � x2 � x � .

4.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 4.pgFind the cubic polynomial f � x ! such that f � 2 !(� 11 � f )*� 2 !+� 14 �f ) ),� 2 !-� 18 � and f ) ) )*� 2 !-� 12 'f � x !$� x3 � x2 � x � .

5.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 5.pgConsider the chemical reaction

aC2H6 � bCO2 � cH2O . dC2H5OH �where a � b � c � and d are unknown positive integers. The reactionmush be balanced; that is, the number of atoms of each elementmust be the same before and after the reaction. For example,because the number of oxygen atoms must remain the same,

2b � c � d 'While there are many possible choices for a � b � c � and d thatbalance the reaction, it is customary to use the smallest possibleintegers. Balance this reaction.a �b �c �d �

6.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 6.pgIn a grid of wires, the temperature at exterior mesh poins ismaintained at constant values as shown in the figure. Whenthe grid is in thermal equilibrium, the temperature at each in-terior mesh point is the average of the temperatures at the fouradjacent points. For instance,

T1 � T2 � T3 � 0 � 204

'Find the temperatures T1 � T2 � T3 � T4 � when the grid is in thermalequilibrium.

T1 �T2 �T3 �T4 �

7.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 7.pgConsider a two-commodity market. When the unit prices of theproducts are P1 and P2, the quantities demanded, D1 and D2, andthe quantities supplied, S1 and S2, are given by

D1 � 202 � 2P1 � P2D2 � 221 � P1 � 3P2

S1 �� 68 � 3P1S2 �� 73 � 2P2

(a) What is the relationship between the two commodities?Do they compete, as do Volvos and BMWs, or do they comple-ment one another, as do shirts and ties? (type in ”compete” or”complement”)

(b) Find the equilibrium prices (i.e. the prices for which sup-ply equals demand), for both products.P1 � P2 �

8.(1 pt) setLinearAlgebra2SystemsApplications/ur la 2 8.pgA dietician is planning a meal that supplies certain quantities ofvitamin C, calcium, and magnesium. Three foods will be used,their quantities measured in millirams. The nutrients suppliedby one unit of each food and the dietary requirements are givenin the table below.

Nutrient Food 1 Food 2 Food 3 Total Required (mg)Vitamin C 40 20 20 260Calcium 20 20 20 200

Magnesium 10 15 15 1351

Write the augmented matrix for this problem. ��What quantity (in units) of Food 1 is neccesary to meet the

dietary requirements?

What quantity (in units) of Food 2 is neccesary to meet thedietary requirements?

What quantity (in units) of Food 3 is neccesary to meet thedietary requirements?


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra3Matrices due 1/3/09 at 2:00 AM

1.(1 pt) setLinearAlgebra3Matrices/ur la 3 5.pgGiven the augmented matrix A , perform each row operation inorder, (a) followed by (b) followed by (c).

A � 1 2 � 4 62 5 3 6� 1 � 3 2 5

�� a ! R2 �� 2r1 � r2� b ! R3 � 1r1 � r3� c ! R3 � 1r2 � r3 ��

2.(1 pt) setLinearAlgebra3Matrices/ur la 3 6.pg

If A and B are 9 / 7 matrices, and C is a 2 / 9 matrix, which ofthe following are defined?� A. BC� B. CA� C. B � A� D. C � B� E. AT� F. ABT

3.(1 pt) setLinearAlgebra3Matrices/ur la 3 13.pgIf A � B � and C are 5 / 5 � 5 / 8 � and 8 / 9 matrices respectively,determine which of the following products are defined. Forthose defined, enter the size of the resulting matrix (e.g. ”3 x 4”,with spaces between numbers and ”x”). For those undefined,enter ”undefined”.

BA:AB:AC:A2:

4.(1 pt) setLinearAlgebra3Matrices/ur la 3 14.pgFind the ranks of the following matrices.

rank

�3 � 76 � 14 � �

rank

�� 0 0 0 5� 9 0 0 00 0 1 00 5 0 0

� �� rank

0 � 20 � 20 8

�� 5.(1 pt) setLinearAlgebra3Matrices/ur la 3 1b.pg

If A � 4 4 � 4� 4 � 1 � 33 1 4

��and B � � 4 � 2 1� 4 � 2 � 1

2 � 1 � 1

��

Then 4A � B � ��6.(1 pt) setLinearAlgebra3Matrices/ur la 3 1.pg

If A � 1 3 11 3 1� 1 � 4 2

��and B �

3 � 3 0� 2 � 3 01 � 3 1

��Then 3A � B � ��and AT � ��

7.(1 pt) setLinearAlgebra3Matrices/ur la 3 1a.pg

If A � � 3 � 4 � 33 � 3 � 41 2 0

��and B �

0 � 2 40 2 43 � 4 � 1

��Then 2A � 4B � ��and 5AT � ��


If A � � � 1 0� 2 � 4 � and B � � � 7 63 2 �

Then AB � � �9.(1 pt) setLinearAlgebra3Matrices/ur la 3 16.pg

Compute the following product.�2 � 40 � 5 � � � 5� 3 � � � �10.(1 pt) setLinearAlgebra3Matrices/ur la 3 17.pg

Compute the following product.�5 � 5 � 21 2 � 5 � � 2

23

�� 11.(1 pt) setLinearAlgebra3Matrices/ur la 3 18.pg

Compute the following product.0 � 1 � 1 2 � 2 1 �� 113� 3

� �� 0 112.(1 pt) setLinearAlgebra3Matrices/ur la 3 8.pg

Compute the following product.6 � 88 7� 6 � 9

�� 52 � � ��

1

13.(1 pt) setLinearAlgebra3Matrices/ur la 3 19.pgCompute the following product.

4 � 2 � 3� 5 � 2 4� 3 3 3

�� 31� 4

�� 14.(1 pt) setLinearAlgebra3Matrices/ur la 3 4.pg

Compute the following product.1 34 � 11 � 3

�� 3 � 3� 3 � 1 � � ��15.(1 pt) setLinearAlgebra3Matrices/ur la 3 12.pg

Compute the following product.0 � 3 � 3 � 2 1 2 � 23 2� 2 � 1

�� 0 116.(1 pt) setLinearAlgebra3Matrices/ur la 3 28.pg

Compute the following products.�2 3 � 31 2 � 1 � � 1 � 1

3 � 13 2

�� 1 � 13 � 13 2

�� 2 3 � 31 2 � 1 � � ��

17.(1 pt) setLinearAlgebra3Matrices/ur la 3 20.pgFind a 3 / 3 matrix A such that

A

100

�� 354

��,

A

010

�� 25� 3

��, and

A

001

�� 554

��.

A � ��


If A � � 3 � 2 � 2� 4 � 3 4� 1 2 � 2

��and B � � 3 3 � 1� 3 3 1� 2 � 2 � 4

��Then AB � ��and BA � ��


If A � � � 1 � 3i � 1� 3 3 � 3i � and B � �2 � 3i 4 � 3i4 � 4i 4 � 3i �

Then AB � � �and BA � � �


If A � 3 � 1 10 � 3 2� 6 5 � 4

��Then rankA � , and

A2 � ��21.(1 pt) setLinearAlgebra3Matrices/ur la 3 26.pg

Find the value of k for which the matrix

A � � 5 � 5 � 151 4 63 � 6 k

��has rank 2.

k �22.(1 pt) setLinearAlgebra3Matrices/ur la 3 15.pg

Find a and b such that � 18� 8� 24

�� a

112

�� b

728

��.

a �b �


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra4InverseMatrix due 1/4/09 at 2:00 AM

1.(1 pt) setLinearAlgebra4InverseMatrix/ur la 4 1.pg

If A � � � 9 8� 2 1 �Then A 2 1 � � �


The matrix

�6 2� 6 k � is invertible if and only if k 3� .


If A � 1 9 � 80 1 � 40 0 1

��Then A 2 1 � ��


If A � 5 15 392 7 191 3 8

��Then A 2 1 � ��

5.(1 pt) setLinearAlgebra4InverseMatrix/ur la 4 5.pgA square matrix is called a permutation matrix if it contains theentry 1 exactly once in each row and in each column, with allother entries being 0. All permutation matrices are invertible.Find the inverse of the following permutation matrix

A � �� 0 0 1 00 0 0 10 1 0 01 0 0 0

��

A 2 1 � �� 6.(1 pt) setLinearAlgebra4InverseMatrix/ur la 4 6.pg

If A � �� 1 0 0 05 1 0 0� 8 � 2 1 0� 7 2 � 5 1

� ��Then A 2 1 � ��


If A � �� 5 � 1 0 0� 16 3 0 00 0 1 � 10 0 � 3 2

� ��Then A 2 1 � ��


Determine which of the formulas are hold for all invertible n / nmatrices A and B� A. � A � B !4� A � B !+� A2 � B2� B. AB � BA� C. A � In is invertible� D. A6 is invertible� E. � AB !52 1 � A 2 1B 2 1� F. � In � A !4� In � A 2 1 !6� 2In � A � A 2 1


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra5LUfactorization due 1/5/09 at 2:00 AM

1.(1 pt) setLinearAlgebra5LUfactorization/ur la 5 1.pg

Find the LU factorization of A � �4 5� 16 � 17 � , i.e. write

A � LU where L is a lower triangular matrix with 1’s on thediagonal, and U is an upper triangular matrix.

A � � � � � .


Find the LU factorization of A � �3 3 � 56 5 � 8 � .

A � � � � � .


Find the LU factorization of A � 5 3� 15 � 8

20 10

��.

A � �� .


Find the LU factorization of A � � 4 1 � 4� 4 6 116 16 41

��.

A � �� .

5.(1 pt) setLinearAlgebra5LUfactorization/ur la 5 5.pgFind the LU factorization of

A � � 4 � 2 � 1 � 28 1 1 14 � 13 � 8 � 10

��.

A � LU where

L � ��,

U � ��.


A � �� 4 3 � 2 2� 20 12 � 13 148 � 3 4 � 3� 8 � 9 � 7 1

��.

A � LU where

L � �� ,

U � �� .


Find the LU factorization of A � �3 4� 12 � 21 � , and use it

to solve the system

�3 4� 12 � 21 � �

x1x2 � � �

18� 87 � .

A � � � � � ,x1 � ,x2 � .


Find the LU factorization of A � � 4 0 � 2� 8 � 4 � 1� 4 � 4 � 2

��,

and use it to solve the system � 4 0 � 2� 8 � 4 � 1� 4 � 4 � 2

�� x1x2x3

�� 61� 2

��.

A � �� ,

x1 � ,x2 � ,x3 � .


A � �� 2 � 1 � 3 � 30 2 � 4 4� 6 9 � 7 230 � 2 8 � 2

� ��,

and use it to solve the system�� 2 � 1 � 3 � 30 2 � 4 4� 6 9 � 7 230 � 2 8 � 2

�� x1x2x3x4

�� 122664� 36

��.

A � LU where

L � �� ,

U � �� ,

x1 � ,x2 � ,x3 � ,x4 � .

1


Find the LDU factorization of A � � � 3 � 15� 3 � 19 � , i.e. write

A � LDU where L is a lower triangular matrix with 1’s on thediagonal, D is a diagonal matrix, and U is an upper triangularmatrix with 1’s on the diagonal.

A � � � � � � � .


Find the LDU factorization of A � 5 � 20 � 5� 25 101 285 � 24 � 12

��.

A � LDU where

L � ��,

D � ��,

U � ��.

12.(1 pt) setLinearAlgebra5LUfactorization/ur la 5 12.pgFind the LDU factorization of

A � �� 1 3 2 4� 3 5 � 6 � 44 4 42 522 6 32 37

��.

A � LDU where

L � �� ,

D � �� ,

U � �� .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra6Determinants due 1/6/09 at 2:00 AM

1.(1 pt) setLinearAlgebra6Determinants/ur la 6 3.pgFind the determinant of the matrix

A � �7 � 25 � 8 �

det � A !-�2.(1 pt) setLinearAlgebra6Determinants/ur la 6 11.pg

Find the determinant of the matrix

M � � 3 � 3 � 50 5 70 0 2

��det � M !$�

3.(1 pt) setLinearAlgebra6Determinants/ur la 6 12.pgA square matrix is called a permutation matrix if each row andeach column contains exactly one entry 1, with all other entriesbeing 0. An example is

P � 0 0 10 1 01 0 0

��Find the determinant of this matrix.det � P !-�


B � � 3 � 5 20 � 3 00 � 2 2

��det � B !-�


A � �� 2 0 0 01 � 5 0 01 � 3 4 0� 7 5 � 9 3

� ��det � A !-�


M � �� 1 0 0 3� 2 0 2 00 � 1 0 � 10 � 2 2 0

� ��det � M !$�


C � �� 1 2 2 13 � 1 2 0� 1 0 0 00 2 2 3

� ��det � C !6�


M �� 2 0 0 � 1 0� 3 0 � 3 0 0

0 � 3 0 0 20 0 0 � 1 � 20 � 3 � 1 0 0

� ��det � M !6�

9.(1 pt) setLinearAlgebra6Determinants/ur la 6 16.pgFind the determinant of the n / n matrix A with 6’s on the diag-onal, 1’s above the diagonal, and 0’s below the diagonal.det � A !-�

10.(1 pt) setLinearAlgebra6Determinants/ur la 6 2.pg

If A � � � 4 6� 9 � 3 � ,

Then det � A !6� and A 2 1 � � �11.(1 pt) setLinearAlgebra6Determinants/ur la 6 8.pg

Find k such that the matrix

M � 4 � 3 14 � 5 5

18 � k � 13 11

��is singular.k �

12.(1 pt) setLinearAlgebra6Determinants/ur la 6 17.pgIf the determinant of a 3 / 3 matrix A is det � A !7� 10, and thematrix B is obtained from A by multiplying the second row by8, then det � B !6� .

13.(1 pt) setLinearAlgebra6Determinants/ur la 6 18.pgIf the determinant of a 4 / 4 matrix A is det � A !8�9� 9, and thematrix C is obtained from A by swapping the second and thirdrows, then det � C !6� .

14.(1 pt) setLinearAlgebra6Determinants/ur la 6 19.pgIf the determinant of a 4 / 4 matrix A is det � A !:� 2, and thematrix D is obtained from A by adding 7 times the fourth row tothe second, then det � D !-� .

15.(1 pt) setLinearAlgebra6Determinants/ur la 6 9.pgIf A and B are 2 / 2 matrices, det � A !-�� 2, det � B !$�� 8, thendet � AB !-�det � 3A !-�det � AT !-�det � B 2 1 !6�det � B2 !6�

16.(1 pt) setLinearAlgebra6Determinants/ur la 6 20.pg

If det

a 1 db 1 ec 1 f

�� 5 and det

a 1 db 2 ec 3 f

�� 5,

Then det

a 9 db 9 ec 9 f

�� ,

1

and det

a � 2 db � 3 ec � 4 f

�� .

17.(1 pt) setLinearAlgebra6Determinants/ur la 6 22.pgSuppose that a 4 / 4 matrix A with rows v1, v2, v3, and v4 hasdeterminant detA �� 4. Find the following determinants deter-minants:

det

�� 9v1v2v3v4

� �� det

�� v2v1v4v3

� �� det

�� v1v2v3

v4 � 3v2

� �� 18.(1 pt) setLinearAlgebra6Determinants/ur la 6 23.pg

If a 4 / 4 matrix A with rows v1, v2, v3, and v4 has determinantdetA � 2,

Then det

�� 7v1 � 3v4v2v3

9v1 � 4v4

� �� 19.(1 pt) setLinearAlgebra6Determinants/ur la 6 21.pg

Find the derivative of the function

f � x !$� det

�� 5 � 5 3 � 1 36 0 � 6 � 8 � 9� 2 0 0 3 � 6x 3 4 � 2 1� 5 0 0 0 � 3

��.

f )*� x !6�20.(1 pt) setLinearAlgebra6Determinants/ur la 6 1.pg

Consider the following general matrix equation:�a1a2 � � �

m11 m12m21 m22 � �

x1x2 �

which can also be abbreviated as:A � MX

By definition, the determinant of M is given by

det � M !6� m11m22 � m12m21

The following questions are about the relationship between thedeterminant of M and the ability to solve the equation above forA in terms of X or for X in terms of A.Check the boxes which make the statement correct:If the det � M !;3� 0 then� A. given any A there is one and only one X which will

satisfy the equation.� B. some values of X will have no values of A whichsatisfy the equation.� C. given any X there is one and only one A which willsatisfy the equation.� D. some values of A will have no values of X which willsatisfy the equation.� E. some values of A (such as A � 0 ) will allow morethan one X to satisfy the equation.� F. some values of X will have more than one value of Awhich satisfy the equation.

Check the boxes which make the statement correct:If the det � M !$� 0 then� A. given any A there is one and only one X which will

satisfy the equation.� B. some values of A (such as A � 0 ) will allow morethan one X to satisfy the equation.� C. some values of A will have no values of X which willsatisfy the equation.� D. there is no value of X which satisfies the equationwhen A � 0.� E. given any X there is one and only one A which willsatisfy the equation.

Check the conditions that guarantee that det � M !$� 0:� A. There is some value of A for which no value of Xsatisfies the equation.� B. When A � 0 there is more than one X which satisfiesthe equation.� C. Given any X there is one and only one A which willsatisfy the equation.� D. Given any A the is one and only one X which willsatisfy the equation.


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra7AreaVolume due 1/7/09 at 2:00 AM

1.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 7.pgFind the area of the parallelogram defined by the vectors�

34 � and

� � 41 � .

Area = .

2.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 1.pgFind the area of the parallelogram definded by the vectors�� 1

222

� ��and

�� 2� 211

� ��.

Area = .

3.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 2.pgFind the area of the parallelogram with vertices at � 4 � 1 !"�� 6 �<� 9 !"�#� � 8 �%� 10 !"� and � � 6 �<� 20 !4'Area = .

4.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 3.pgFind the area of the triangle with vertices � � 4 � 2 ! , � 4 � 3 ! , and� � 7 � 7 ! .Area = .

5.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 4.pgFind the area of the quadrangle with vertices � 6 � 2 ! , � � 5 � 4 ! ,� � 3 �<� 5 ! , and � 4 �%� 3 ! .

Area = .6.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 8.pg

Find the volume of the parallelepiped defined by the vectors � 324

��,

� 3� 10

��, and

413

��.

Volume = .7.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 5.pg

Find the volume of the parallelepiped definded by the vectors�� 2123

� ��,

�� 1� 122

� ��, and

�� 312� 1

� ��.

Volume = .8.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 6.pg

Find the volume of the parallelepiped with one vertex at�%� 3 �<� 2 � 2 !"� and adjacent vertices at � � 3 � 0 � 2 !4�=� 0 � 1 � 2 !4� and�%� 5 � 0 � 9 !"'Volume = .

9.(1 pt) setLinearAlgebra7AreaVolume/ur la 7 9.pgFind the volume of the tetrahedron with vertices � 1 � 1 �%� 5 ! ,�%� 2 � 0 �%� 2 ! , � 6 � 4 �<� 5 ! , and � 2 �%� 1 �%� 10 ! .Volume = .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra8VectorSpaces due 1/8/09 at 2:00 AM

1.(1 pt) setLinearAlgebra8VectorSpaces/ur la 8 1.pgWhich of the following subsets of P2 are subspaces of P2?� A. > p � t !@? p )*� t !A� 2p � t !A� 4 � 0 B� B. > p � t !@?DC 6

0 p � t ! dt � 0 B� C. > p � t !@? p � 2 !-� 5 B� D. > p � t !@? p � 4 !-� 0 B� E. > p � t !:? p �%� t !E�� p � t ! for all t B� F. > p � t !=? p )F� t ! is constant B2.(1 pt) setLinearAlgebra8VectorSpaces/ur la 8 2.pg

Which of the following subsets of G 3 H 3 are subspaces of G 3 H 3 ?� A. The 3 / 3 matrices with trace 0 (the trace of a matrixis the sum of its diagonal entries)� B. The invertible 3 / 3 matrices� C. The diagonal 3 / 3 matrices� D. The 3 / 3 matrices in reduced row-echelon form� E. The 3 / 3 matrices whose entries are all integers� F. The 3 / 3 matrices with all zeros in the third row

3.(1 pt) setLinearAlgebra8VectorSpaces/ur la 8 3.pgDetermine whether the given set S is a subspace of the vectorspace V .� A. V � C5 � I ! , and S is the subset of V consisting

of those functions satisfying the differential equationy I 5 J � 0 '

� B. V � P5, and S is the subset of P5 consisting of thosepolynomials satisfying p � 1 !LK p � 0 !4'� C. V � Mn �*G8! , and S is the subset of all symmetric ma-trices� D. V �MG 5 , and S is the set of vectors � x1 � x2 � x3 ! in Vsatisfying x1 � 6x2 � x3 � 5 '� E. V � Pn, and S is the subset of Pn consisting of thosepolynomials satisfying p � 0 !-� 0 '� F. V � C3 � I ! , and S is the subset of V consisting of thosefunctions satisfying the differential equation y ) ) )"� 4y �x2 '� G. V is the vector space of all real-valued functions de-fined on the interval � � ∞ � ∞ ! , and S is the subset of Vconsisting of those functions satisfying f � 0 !6� 0 '

4.(1 pt) setLinearAlgebra8VectorSpaces/ur la 8 4.pg(a) If S is the subspace of M6 �FG7! consisting of all symmetricmatrices, then dim S �(b) If S is the subspace of M8 �FG7! consisting of all skew-symmetric matrices, then dim S �

5.(1 pt) setLinearAlgebra8VectorSpaces/ur la 8 5.pgFind the dimensions of the following linear spaces.(a) P6(b) The space of all diagonal 3 / 3 matrices(c) G 4 H 7


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra9Dependence due 1/9/09 at 2:00 AM

1.(1 pt) setLinearAlgebra9Dependence/ur la 9 1.pg

Let A � � 331

��, B �

6� 9� 5

��, and C �

9� 12� 2

��.

? 1. Determine whether or not the three vectors listed aboveare linearly independent or linearly dependent.

If they are linearly dependent, determine a non-trivial linear re-lation - (a non-trivial relation is three numbers which are notall three zero.) otherwise, if the vectors are linearly indepen-dent, enter 0’s for the coefficients, since that relationship alwaysholds.

A � B � C � 0.You can use this row reduction tool to help with the calcula-tions.

2.(1 pt) setLinearAlgebra9Dependence/ur la 9 2.pgLet A � 0

4 3 5 1 , B � 04 3 4 1 ,

and C � 0 � 4 � 2 � 1 1 .

? 1. Determine whether or not the three vectors listed aboveare linearly independent or linearly dependent.

The vectors were written horizontally this time, as they often arein books, but that is just to save space. The problem is the sameas if the vectors were written vertically.If they are linearly dependent, determine a non-trivial linear re-lation - (a non-trivial relation is three numbers which are notall three zero.) otherwise, if the vectors are linearly indepen-dent, enter 0’s for the coefficients, since that relationship alwaysholds.

A � B � C � 0.You can use this row reduction tool to help with the calcula-tions.


Let A � �� 3� 30� 4

� ��, B � �� 9

10414

� ��, C � �� 3

436

� ��, and

D � �� 3313

� ��.

? 1. Determine whether or not the four vectors listed aboveare linearly independent or linearly dependent.

If they are linearly dependent, determine a non-trivial linear re-lation - (a non-trivial relation is three numbers which are notall three zero.) Otherwise, if the vectors are linearly indepen-dent, enter 0’s for the coefficients, since that relationship alwaysholds.

A � B � C � D � 0.You can use this row reduction tool to help with the calcula-tions.

4.(1 pt) setLinearAlgebra9Dependence/ur la 9 4.pgLet f � exp � t ! , g � t, and h � 2 � 3 N t. Give the answer 1 if f �g � and h are linearly dependent and 0 if they are linearly inde-pendent.linearly dependent? �

5.(1 pt) setLinearAlgebra9Dependence/ur la 9 5.pgDetermine which of the following pairs of functions are linearlyindependent.

? 1. f � t !L� 4t2 � 28t � g � t !-� 4t2 � 28t? 2. f � θ !L� cos � 3θ !O� g � θ !6� 4cos3 � θ !P� 8cos � θ !? 3. f � x !$� e4x � g � x !6� e4 I x 2 3 J? 4. f � t !L� eλt cos � µt !O� g � t !-� eλt sin � µt !O� µ 3� 0

6.(1 pt) setLinearAlgebra9Dependence/ur la 9 6.pgDetermine whether the following pairs of functions are linearlyindependent or not.

? 1. f � t !L� t and g � t !-�Q? t ?? 2. f � x !$� e6x and g � x !-� e6 I x 2 1 J? 3. f � θ !L� 6cos3θ and g � θ !6� 24cos3 θ � 18cosθ


The vectors

v � 5� 19� 29

��, u � � 2

64 � k

��, and w �

1� 7� 9

��.

are linearly independent if and only if k 3� .


Let v1 � 35� 3

��, v2 � � 6� 8

10

��, and y �

2127h

��.

For what value of h is y in the plane spanned by v1 and v2?h �


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra10Bases due 1/10/09 at 2:00 AM

1.(1 pt) setLinearAlgebra10Bases/ur la 10 1.pgThe vectors

v1 � 7� 70

��, v2 � � 7

37

��, and v3 � � 7� 9

k

��form a basis for G 3 if and only if k 3� .

2.(1 pt) setLinearAlgebra10Bases/ur la 10 2.pgConsider the basis B of G 2 consisting of vectors� � 7

3 � and

� � 1� 4 � .

Find x in G 2 whose coordinate vector relative to the basis B isRx S B � �

4� 1 � .

x � � �3.(1 pt) setLinearAlgebra10Bases/ur la 10 3.pg

The set B � � �4� 6 � � � � 12

20 �UT is a basis for G 2 .

Find the coordinates of the vector x � �16� 28 � relative to the

basis B:Rx S B � � �

4.(1 pt) setLinearAlgebra10Bases/ur la 10 4.pg

Find the coordinate vector of x � 5� 4� 4

��with respect to the

basis B � �� 177

�� 01� 4

�� 001

��WV XY or G 3 .Rx S B � ��

5.(1 pt) setLinearAlgebra10Bases/ur la 10 5.pgLet B be the basis of G 2 consisting of the vectors�

43 � and

� � 15 � ,

and let R be the basis consisting of�2� 3 � and

� � 12 � .

Find a matrix P such thatRx S R � P

Rx S B for all x in G 2 .

P � � �6.(1 pt) setLinearAlgebra10Bases/ur la 10 6.pg

The set B �W> 3 � 4x2 � 6 � 1x � 8x2 �-� 19 � 3x � 28x2 B is a basisfor P2. Find the coordinates of p � x !E� 11 � 2x � 20x2 relative tothis basis:Rp � x !ZS B � ��


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra11Eigenvalues due 1/11/09 at 2:00 AM

1.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 9.pgSupppose A is an invertible n / n matrix and v is an eigenvectorof A with associated eigenvalue � 3. Convince yourself that v isan eigenvalue of the following matrices, and find the associatedeigenvalues:1. A6 � eigenvalue = ,2. A 2 1 � eigenvalue = ,3. A � 4In � eigenvalue = ,4. � 9A � eigenvalue = .

2.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 1.pgFind the characteristic polynomial of the matrix

A � � � 8 10� 2 1 � .

p � x !6� .

3.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 2.pgFind the characteristic polynomial of the matrix

A � � 3 1 00 � 2 � 2� 1 � 5 0

��.

p � x !6� .

4.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 3.pgFind the eigenvalues of the matrix

A � � � 19 16� 24 21 � .

The smaller eigenvalue is λ1 � .The bigger eigenvalue is λ2 � .

5.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 4.pgThe matrix

C � 39 � 2 82� 8 � 1 � 16� 18 1 � 38

��has three distinct eigenvalues, λ1 [ λ2 [ λ3, where λ1 � ,λ2 � , and λ3 � .


C � 3 5 50 3 00 � 5 � 2

��has two distinct eigenvalues, λ1 [ λ2:λ1 � has multiplicity , andλ2 � has multiplicity .

7.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 6.pgCalculate the eigenvalues of this matrix:[Note– you may want to use a graphing calculator to estimatethe roots of the polynomial which defines the eigenvalues. Youcan use the web version at xFunctions ]

A � �27 36� 18 � 27 � . smaller eigenvalue � ,

associated eigenvector � ( , ),larger eigenvalue � ,associated eigenvector � ( , ).


C � 25 20 38� 11 � 9 � 17� 12 � 10 � 18

��has two distinct eigenvalues, λ1 [ λ2:λ1 � has multiplicity . The dimension of thecorresponding eigenspace is .λ2 � has multiplicity . The dimension of thecorresponding eigenspace is .Is the matrix C diagonalizable? (enter YES or NO)

9.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 8.pgCalculate the eigenvalues of this matrix:[Note– you may want to use a graphing calculator to estimatethe roots of the polynomial which defines the eigenvalues. Youcan use the web version at xFunctions . Also, You can use thisrow reduction tool to help with the calculations.]

A � � 14 20 � 180 0 09 � 10 13

��.

The eigenvalues are λ1 [ λ2 [ λ3, whereλ1 � ,associated eigenvector � ( , , ),λ2 � ,associated eigenvector � ( , , ),λ3 � ,associated eigenvector � ( , , ).

10.(1 pt) setLinearAlgebra11Eigenvalues/ur la 11 10.pgFind a 2 / 2 matrix A for which

E 2 1 � span

�25 � and E1 � span

�13 �

where Eλ is the eigenspace associated with the eigenvalue λ.

A � � � .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra12Diagonalization due 1/12/09 at 2:00 AM

1.(1 pt) setLinearAlgebra12Diagonalization/ur la 12 1.pg

Let M � �5 4� 2 11 � .

Find formulas for the entries of Mn, where n is a positive integer.

Mn � � � .


Let M � �8 8� 4 � 4 � .


Mn � � � .


Let M � �14 1� 36 2 � .


Mn � � � .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra13ComplexEigenvalues due 1/13/09 at 2:00 AM

1.(1 pt) setLinearAlgebra13ComplexEigenvalues/ur la 13 1.pgThe matrix

A � � � 1 � 68 � 2 �

has complex eigenvalues, λ1 \ 2 � a ] bi, where a � andb � .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra14TransfOfRn due 1/14/09 at 2:00 AM

1.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 9.pg

Which of the following transformations are linear?� A.

�� y1 � 2x2y2 �� 8x3y3 �� 7x1� B.

�� y1 � 6y2 � 5y3 � 2� C.

�y1 � 0y2 � 5x2� D.

�� y1 � x22

y2 � x3y3 � x1� E.

�� y1 �� 3x1y2 � 10x1y3 � 8x1� F.

�y1 � 7x1y2 � 9

2.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 1.pgIf T : G 3 .^G 3 is a linear transformation such that

T

100

�� 401

��,T

010

�� 14� 1

��,

and T

001

�� 40� 4

��,

then T

� 4� 34

�� .


T

�10 � � � � 7� 7 � and T

�01 � � � � 9� 6 � ,

then the standard matrix of T is A � � � .

4.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 10.pgConsider a linear transformation T from G 3 to G 2 for which

T

100

�� 09 � , T

010

�� 52 � ,

and T

001

�� 63 � .

Find the matrix A of T .

A � � �

5.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 12.pgFind the matrix A of the linear transformation from G 2 to G 3

given by

T

�x1x2 � �

631

��x1 �

22� 2

��x2.

A � ��6.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 4.pg

If T : G 2 .^G 2 is a linear transformation such that

T

�16 � � � � 12

33 � and T

�6� 5 � � �

51� 7 � ,

then the standard matrix of T is A � � � .


T

�14 � �

119� 7

��and T

�4� 1 � �

21� 96

��,

then the standard matrix of T is A � ��.


Let b1 � �1� 1 � and b2 � �

2� 1 � .

The set B ��> b1 � b2 B is a basis for G 2 'Let T : G 2 ._G 2 is a linear transformation such thatT � b1 !$� 5b1 � 5b2 and T � b2 !6� 6b1 � 7b2 'Then the matrix of T relative to the basis B isRT S B � � �

and the matrix of T relative to the standard basis E for G 2 isRT S E � � �

9.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 6.pgA linear transformation T : G 3 .`G 2 whose matrix is� � 3 3 6� 2 2 � 5 � k �is onto if and only if k 3� .

10.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 7.pgThe matrix

A � 1 � 2 � 4 � 2

1 ' 5 0 � 6 � 37 ' 5 � 6 � 30 � 11

��is a matrix of a linear transformation T : G k .^G n wherek � , n � ,dim(Ker � T !%!6� , dim(Range � T !<!-� .Is T onto? (enter YES or NO)Is T one-to-one? (enter YES or NO)

1


Let A � � 4 86 � 13� 3 9

��and b � � 48

77� 51

��.

A linear transformation T : G 2 ._G 3 is defined by T � x !$� Ax.

Find an x � �x1x2 � in G 2 whose image under T is b.

x1 �x2 �

12.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 13.pgMatch each linear transformation with its matrix.

1.

� � 1 00 1 �

2.

�1 00 0 �

3.

� � 1 00 � 1 �

4.

�1 00 1 �

5.

�1 00 � 1 �

6.

�2 00 2 �

A. Reflection in the y-axisB. Projection onto the x-axisC. Reflection in the originD. Reflection in the x-axisE. Dilation by a factor of 2F. Identity transformation

13.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 14.pgFind a 3 / 3 matrix A such that Ax � 7x for all x in G 3 .


Find the matrix A of the linear transformation T from G 2 to G 2

that rotates any vector through an angle of 120 a in the counter-clockwise direction.

A � � �15.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 16.pg

The dot product of two vectors in G 3 is defined bya1a2a3

��cb b1b2b3

�� a1b1 � a2b2 � a3b3 'Let v �

490

��. Find the matrix A of the linear transformation

from G 3 to G given by T � x !6� v

bx '

A � 0 1

16.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 17.pgThe cross product of two vectors in G 3 is defined by

a1a2a3

�� / b1b2b3

�� a2b3 � a3b2a3b1 � a1b3a1b2 � a2b1

��.

Let v � � 21� 7

��. Find the matrix A of the linear transforma-

tion from G 3 to G 3 given by T � x !6� v / x 'A � ��

17.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 18.pgLet L be the line in G 3 that consists of all scalar multiples of the

vector

1� 2� 2

��. Find the orthogonal projection of the vector

v � 484

��onto L.

projLv � ��18.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 19.pg

Let L be the line in G 3 that consists of all scalar multiples of the

vector

22� 1

��. Find the reflection of the vector v �

284

��in the line L.

projLv � ��19.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 20.pg

Find the matrix A of the orthogonal projection onto the line L inG 2 that consists of all scalar multiples of the vector

�53 � .


Find the matrix A of the reflection in the line L in G 2 that con-

sists of all scalar multiples of the vector

�53 � .


Find the matrices of the following linear transformations fromG 3 to G 3 .The orthogonal projection onto the yz-plane: ��The reflection in the x-axis:

2

��22.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 23.pg

Find the matrix A of the rotation about the x -axis through an an-gle of π

2 , counterclockwise as viewed from the positive x -axis.


Let A � �8 20 9 � and B � �

6 71 5 � .

Find the matrix C of the linear transformation T � x !-� B � A � x !%!"'C � � �


Which of the following linear transformations from G 3 to G 3

are invertible?� A. Reflection in the y -axis� B. Identity transformation (i.e. T � v !6� v for all v)� C. Rotation about the x -axis� D. Projection onto the z -axis

� E. Trivial transformation (i.e. T � v !-� 0 for all v)� F. Dilation by a factor of 4

25.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 11.pgFind the inverse of the linear transformation

y1 � 2x1 � 3x2y2 �� 3x1 � 5x2

x1 � y1 � y2x2 � y1 � y2

26.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 25.pgFind the inverse of the linear transformation

y1 � 4x1 � 8x2 � 21x3y2 � 3x1 � 7x2 � 16x3y3 � x1 � 2x2 � 5x3

x1 � y1 � y2 � y3x2 � y1 � y2 � y3x3 � y1 � y2 � y3

27.(1 pt) setLinearAlgebra14TransfOfRn/ur la 14 26.pgFind the inverse of the (nonlinear) transformation from G 2 to G 2

given byu � 6yv � 9x7 � 8y

x �y �


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra15TransfOfLinSpaces due 1/15/09 at 2:00 PM

1.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 1.pgWhich of the following transformatiions are linear?� A. T � A !-� A2 from G 4 H 4 to G 4 H 4� B. T � A !d� SAS 2 1 from G 2 H 2 to G 2 H 2 , where

S � �7 3� 1 0 �� C. T � A !d� ASA 2 1 from G 2 H 2 to G 2 H 2 , where

S � � � 3 94 � 6 �� D. T � A !-� 9A from G 4 H 3 to G 4 H 3� E. T � A !6� trace � A ! from G 6 H 6 to G� F. T � A !$� AT from G 5 H 3 to G 3 H 5

2.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 2.pgWhich of the following transformations are linear?� A. T � f � t !%!6�fe 42 6

f � t ! dt from P7 to G� B. T � f � t !%!$� f �%� t ! from P2 to P2� C. T � f � t !%!$� f � t ! f ) � t ! from P6 to P11� D. T � f � t !%!6� f ) � t !#� 4 f � t !A� 7 from C∞ to C∞� E. T � f � t !%!$� t8 f ) � t ! from P2 to P9� F. T � x � iy !-� 8x � iy from g to g3.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 3.pg

If T : P1 . P1 is a linear transformation such thatT � 1 � 3x !(�� 2 � 2x and T � 4 � 11x !D� 1 � 2x � thenT � � 4 � 2x !(� .

4.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 4.pg

The matrices A1 � �1 00 0 � , A2 � �

0 10 0 � ,

A3 � �0 01 0 � , and A4 � �

0 00 1 �

form a basis for the linear space V �hG 2 H 2 ' Write the ma-trix of the linear transformation T : G 2 H 2 .�G 2 H 2 such thatT � A !-� 4A � 15AT relative to this basis:��

5.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 5.pgFind the matrix A of the linear transformation

T � M !6� �2 50 8 � M

from U2 H 2 to U2 H 2, with respect to the basis� �1 00 0 � � � 1 1

0 0 � � � 0 00 1 �UT .

A � ��

6.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 6.pgFind the matrix A of the linear transformation

T � M !6� �2 90 4 � M

�2 90 4 � 2 1

from U2 H 2 to U2 H 2, with respect to the standard basis for U 2 H 2:� �1 00 0 � � � 0 1

0 0 � � � 0 00 1 �UT .

A � ��7.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 7.pg

Find the matrix A of the linear transformation T � f � t !%!��9 f )F� t !P� 10 f � t ! from P2 to P2 with respect to the standard ba-sis for P2, > 1 � t � t2 B .


Find the matrix A of the linear transformation T � f � t !%!i� f � 6 !from P2 to P2 with respect to the standard basis for P2, > 1 � t � t2 B .


Find the matrix A of the linear transformation T � f � t !%!6� f � 9t �5 ! from P2 to P2 with respect to the standard basis for P2,> 1 � t � t2 B .


Find the matrix A of the linear transformation

T � f � t !%!6� e 32 5f � t ! dt

from P3 to G with respect to the standard bases for P3 and G .A � 0 1

11.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 11.pgFind the matrix A of the linear transformation T � z !8�j� 9 � 6i ! zfrom g to g with respect to the standard basis g , > 1 � i B .

A � � �12.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 12.pg

Find the matrix A of the linear transformation T � z !8�j� 4 � 8i ! zfrom g to g with respect to the basis > 3 � 2i � 2 � 3i B .

A � � �1

13.(1 pt) setLinearAlgebra15TransfOfLinSpaces/ur la 15 13.pgLet V be the space spanned by the two functions cos � t ! andsin � t ! . Find the matrix A of the linear transformation T � f � t !%!6�f ) ),� t !k� 5 f )l� t !k� 8 f � t ! from V into itself with respect to the basis> cos � t !"� sin � t !"B .


Let V be the plane with equation x1 � 2x2 � 3x3 � 0 in G 3 .Find the matrix A of the orthogonal projection onto the line

spanned by the vector v � 03� 2

��with respect to the basis��

210

�� 301

�� V XY .


Let V be the plane with equation x1 � 3x2 � 4x3 � 0in G 3 . Find the matrix A of the linear transforma-

tion T � x !m� 5 � 1 1� 1 1 � 1� 2 1 � 1

��x with respect to the basis��

310

�� 401

�� V XY .

A � � �


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra16DeterminantOfTransf due 1/16/09 at 2:00 AM

1.(1 pt) setLinearAlgebra16DeterminantOfTransf/ur la 16 1.pgConsider a linear transformation T � x !D� Ax from G 2 to G 2 . Sup-pose for two vectors v1 and v2 in G 2 we have T � v1 !L� 7v2 andT � v2 !-�� 7v1. Find the determinant of the matrix A.det � A !-�

2.(1 pt) setLinearAlgebra16DeterminantOfTransf/ur la 16 2.pgFind the determinant of the linear transformationT � f !6� 5 f � 9 f ) from P2 to P2.det �

3.(1 pt) setLinearAlgebra16DeterminantOfTransf/ur la 16 3.pgFind the determinant of the linear transformationT � f � t !%!6� f � 2t !+� 6 f � t ! from P2 to P2.det �

4.(1 pt) setLinearAlgebra16DeterminantOfTransf/ur la 16 4.pgFind the determinant of the linear transformationT � z !6�Q� 6 � 7i ! z from g to g .det �

5.(1 pt) setLinearAlgebra16DeterminantOfTransf/ur la 16 5.pgFind the determinant of the linear transformation

T � M !6� �4 � 60 6 � M

from the space V of upper triangular 2 / 2 matrices to V .det �


T � M !6� �4 33 � 1 � M � M

�4 33 � 1 �

from the space V of symmetric 2 / 2 matrices to V .det �

7.(1 pt) setLinearAlgebra16DeterminantOfTransf/ur la 16 7.pgFind the determinant of the linear transformationT � f !-� 3 f � 8 f )5� 4 f ) ) from the space V spanned by cos � x ! andsin � x ! to V .det �


T � v !-� 1� 22

�� / v

from the plane E given by x � 2y � 2z � 0 to E.det �


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra17DotProductRn due 1/17/09 at 2:00 AM

1.(1 pt) setLinearAlgebra17DotProductRn/ur la 17 8.pg

Let x � �18 � and y � �

03 � .

Find the dot product of x and y.x

by � .


Let x � 34� 2

��and y � � 1� 1

2

��.

Find the dot product of x and y.x

by � .


Find the length of the vector x � �75 � .?n? x ?o?p� .


Find the length of the vector x � 54� 3

��.?n? x ?o?p� .


Let x � �� 3� 3� 54

��.

Find the norm of x and the unit vector in the direction of x.?n? x ?o?p� ,

u � �� .

6.(1 pt) setLinearAlgebra17DotProductRn/ur la 17 11.pgLet > e1 � e2 � e3 � e4 � e5 � e6 B be the standard basis in G 6 . Find thelength of the vector x �� 5e1 � 3e2 � 4e3 � 2e4 � 2e5 � 5e6.?n? x ?o?p� .

7.(1 pt) setLinearAlgebra17DotProductRn/ur la 17 7.pgFind the value of k for which the vectors

x � �� 20� 1� 4

� ��and y � �� 3

2� 5k

� ��are orthogonal.

k � .


Find the angle α between the vectors

�3� 3 � and

�51 � .

α � .


Find the angle α between the vectors

114

��and

1� 43

��.

α � .10.(1 pt) setLinearAlgebra17DotProductRn/ur la 17 15.pg

Find the orthogonal projection of v � 5� 29

��onto the sub-

space V of G 3 spanned by

� 22� 2

��and

2� 6� 8

��.

projV � v !6� ��.


Find the orthogonal projection of v � �� 20� 211� 11

� ��onto the sub-

space V of G 3 spanned by

�� 4413

� ��and

�� 23� 200

� ��.

projV � v !6� �� .


Find the orthogonal projection of v � �� 0� 300

��onto the subspace V of G 3 spanned by�� 1

1� 1� 1

� ��,

�� 111� 1

� ��, and

�� 11� 11

� ��.

projV � v !6� �� .


Let W be the subspace of G 3 spanned by the vectors

112

��and

2� 12� 10

��. Find the matrix A of the orthogonal projection

onto W .1

A � ��.


Let W be the subspace of G 3 spanned by the vectors

�� 11� 1� 1

� ��and

�� 1� 524

� ��. Find the matrix A of the orthogonal projection

onto W .

A � �� .


Among all the unit vectors u � xyz

��in G 3 ,

find the one for which the sum x � 6y � 4z is minimal.

u � ��.


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra18OrthogonalBases due 1/18/09 at 2:00 AM

1.(1 pt) setLinearAlgebra18OrthogonalBases/ur la 18 1.pgFind the missing coordinates such that the three vectors form anorthonormal basis for G 3 :

0 ' 6� 0 ' 8 ��,

� 1

��,

� 0 ' 6 ��.

2.(1 pt) setLinearAlgebra18OrthogonalBases/ur la 18 2.pg

Let x � 1� 23

��and y � � 4

3� 6

��.

Use the Gram-Schmidt process to determine an orthonormal ba-sis for the subspace of G 3 spanned by x and y. ��

,

��.

3.(1 pt) setLinearAlgebra18OrthogonalBases/ur la 18 3.pgPerform the Gram-Schmidt process on the following sequenceof vectors.

x � 244

��, y � � 4� 2� 5

��, z �

0� 90

��. ��

,

��,

��.


Let x � �� 32� 30

� ��and y � �� 15

436

� ��.

Use the Gram-Schmidt process to determine an orthonormal ba-sis for the subspace of G 4 spanned by x and y.��

,

�� .


Let x � �� 2330

��, y � �� 6

4 ' 52 ' 51

��, and z � �� 8 ' 5� 3 ' 5

5 ' 529 ' 5

��.

Use the Gram-Schmidt process to determine an orthonormal ba-sis for the subspace of G 4 spanned by x, y, and z.��

,

�� ,

�� .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra19QRfactorization due 1/19/09 at 2:00 AM

1.(1 pt) setLinearAlgebra19QRfactorization/ur la 19 1.pg

Find the QR factorization of M � � 666

��.

M � �� 0 1 .


Find the QR factorization of M � 4 � 26 1112 36

��.

M � �� .


Find the QR factorization of M � � 2 � 9 7� 4 � 9 � 10� 4 0 � 7

��.

M � �� .


Find the QR factorization of M � �� 1 3� 1 1� 1 31 � 1

� ��.

M � �� .


Find the QR factorization of M � �� 2 � 4 32 � 4 12 2 � 32 2 � 5

� ��.

M � �� .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra20LeastSquares due 1/20/09 at 2:00 AM

1.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 1.pgFind the least-squares solution x q of the system

1 00 10 0

��x � � 2

67

��.

x q � � � .


2 � 1� 2 15 4

��x � � 4

12� 7

��.

x q8� � � .


4� 5� 2

��x �

192

��.

x q8� 0 1 .

4.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 4.pgFind the least-squares solution x q of the system�� 1 � 1 1

1 1 11 � 1 � 11 1 � 1

� ��x � �� 7

9� 15

� ��.

x q8� ��.

5.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 5.pgFit a linear function of the form f � t !(� c0 � c1t to the data points� � 8 � 53 ! , � 0 �<� 6 ! , � 8 �<� 59 ! , using least squares.c0 � ,c1 � .

6.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 6.pgFit a quadratic function of the form f � t !-� c0 � c1t � c2t2 to thedata points � 0 � 4 ! , � 1 � 1 ! , � 2 � 6 ! , � 3 �<� 1 ! , using least squares.c0 � ,c1 � ,c2 � .

7.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 7.pgFit a trigonometric function of the form f � t !$� c0 � c1 sin � t !r�c2 cos � t ! to the data points � 0 � 2 ' 5 ! , � π

2 � 3 ' 5 ! , � π � 10 ' 5 ! , � 3π2 � 7 ' 5 ! ,

using least squares.c0 � ,c1 � ,c2 � .

8.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 8.pgLet S � t ! be the number of daylight hours on the tth day of theyear in Manley Hot Springs. We are given the following datafor S � t ! :

Day t S � t !January 7 7 5March 7 66 12May 3 123 19July 25 206 20

We wish to fit a trigonometric function of the form

f � t !$� a � bsin

�2π365

t � � ccos

�2π365

t �to these data. Find the best approximation of this form, usingleast squares.a � ,b � ,c � .

9.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 9.pgThe table below lists the height h (in cm), the age a (in years),the gender g (1=”Male”, 0=”Female”), and the weight w (in kg)of some college students.

Height Age Gender Weight166 21 0 65183 22 1 89177 20 1 84171 19 1 79158 21 0 57

We wish to fit a linear function of the form

f � t !$� c0 � c1h � c2a � c3g

to these data. Find the best approximation of this form, usingleast squares.c0 � ,c1 � ,c2 � ,c3 � .

10.(1 pt) setLinearAlgebra20LeastSquares/ur la 20 10.pgDuring the summer months Terry makes and sells necklaces onthe beach. Terry notices that if he lowers the price, he can sellmore necklaces, and if he raises the price than he sells fewernecklaces. The table below shows how the number n of neck-laces sold in one day depends on the price p (in dollars).

Price Number of necklaces sold7 32

12 2415 10

1

(a) Find a linear function of the form

n � c0 � c1 p

that best fits these data, using least squares.c0 � ,c1 � .(b) Find the revenue (number of items sold times the price ofeach item) as a function of price p.

R � .(c) If the material for each necklace costs Terry 5 dollars, findthe profit (revenue minus cost of the material) as a function ofprice p.P � .(d) Finally, find the price that will maximize the profit.p � .


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra21InnerProductSpaces due 1/21/09 at 2:00 AM

1.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 1.pg

Find the norm ?o? x ?n? of x �hs 1 �%� 17 � 1

49 �%� 1343 � 1

2401 �%'<'%'5� 1In2 7 J n t 1 �%'<'%'vuin l2.?n? x ?o?p� .


If A � �a11 a12a21 a22 � and B � �

b11 b12b21 b22 �

are arbitrary vectors in M2 �FGi! , then the mapping[ A � B Kw� a11b11 � a12b12 � a21b21 � a22b22defines an inner product in M2 �FG7! .Use this inner product to determine [ A � B K , ?o? A ?o? , ?n? B ?o? , andthe angle αA \ B between A and B for

A � �4 � 12 1 � and B � � � 5 � 1� 5 � 3 � .[ A � B Kw� ,?n? A ?o?�� ,?n? B ?o?�� ,

αA \ B � .

3.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 3.pgIf A and B are arbitrary m / n matrices, then the mapping[ A � B Kw� trace � AT B ! defines an inner product in G m H n .Use this inner product to find [ A � B K , the norms ?o? A ?n? and ?n? B ?o? ,and the angle αA \ B between A and B for

A � � 3 33 21 � 1

��and B � � 3 3

3 � 12 � 2

��.[ A � B Kw� ,?n? A ?o?�� ,?n? B ?o?�� ,

αA \ B � .

4.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 4.pgIf f � x ! and g � x ! are arbitrary polynomials of degree at most 2,then the mapping[ f � g Kw� f � � 2 ! g �%� 2 !A� f � 0 ! g � 0 !#� f � 1 ! g � 1 !defines an inner product in P2.Use this inner product to find [ f � g K , ?n? f ?o? , ?n? g ?o? , and the angleα f \ g between f � x ! and g � x ! forf � x !$� 4x2 � 2x � 9 and g � x !-� 3x2 � 3x � 1.[ f � g Kw� ,?n? f ?o?p� ,?n? g ?o?�� ,α f \ g � .


Use the inner product [ f � g Kw� e 1

0f � x ! g � x ! dx in the vector

space C0 R 0 � 1 S to find [ f � g K , ?n? f ?o? , ?o? g ?n? , and the angle α f \ gbetween f � x ! and g � x ! for f � x !L� 10x2 � 3 and g � x !$� 3x � 10.[ f � g Kw� ,

?o? f ?o?p� ,?o? g ?o?�� ,α f \ g .

6.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 6.pgUse the inner product[ f � g Kw� f � � 1 ! g � � 1 !#� f � 0 ! g � 0 !A� f � 2 ! g � 2 !in P2 to find the orthogonal projection of f � x !:� 2x2 � 6x � 3onto the line L spanned by g � x !-� 2x2 � 2x � 5.projL � f !$� .


Use the inner product [ f � g Kw� e 1

0f � x ! g � x ! dx in the vector

space C0 R 0 � 1 S to find the orthogonal projection of f � x !D� 5x2 � 4onto the subspace V spanned by g � x !-� x and h � x !6� 1.(Caution: x and 1 do not form an orthogonal basis of V .)projV � f !L� .

8.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 8.pgFind the Fourier coefficients of f � x !:�x� 5x � 5, i.e. numbersa0, bk, ck (note that bk and ck may depend on k ) such thatf � x !8� a0

1y2� b1 sin � x !r� c1 cos � x !r� b2 sin � 2x !r� c2 cos � 2x !r�'<'%' .

a0 � ,bk � if k is odd, and if k is even,ck � if k is odd, and if k is even.


Let M1 � � � 1 � 1� 1 1 � and M2 � �3 � 20 1 � .

Consider the inner product [ A � B Kw� trace � AT B ! in the vec-tor space G 2 H 2 of 2 / 2 matrices. Use the Gram-Schmidt pro-cess to determine an orthonormal basis for the subspace of G 2 H 2

spanned by the matrices M1 and M2.� � ,

� � .

10.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 10.pgLet f � x !$� 8, g � x !6� 2x � 3, and h � x !6�� 9x2 � 5x � 7.Consider the inner product [ p � x !"� q � x !zKw� p �%� 1 ! q � � 1 !E�p � 0 ! q � 0 !P� p � 1 ! q � 1 ! in the vector space P2 of polynomials ofdegree at most 2. Use the Gram-Schmidt process to determinean orthonormal basis for the subspace of P2 spanned by the poly-nomials f � x ! , g � x ! , and h � x ! .

, , ,

11.(1 pt) setLinearAlgebra21InnerProductSpaces/ur la 21 11.pgLet f � x !$� 2, g � x !6� 7x � 8, and h � x !6�� 2x2.

Consider the inner product [ p � x !"� q � x !=Kw� e 4

0p � x ! g � x ! dx in

the vector space C0 R 0 � 1 S . Use the Gram-Schmidt process todetermine an orthonormal basis for the subspace of C0 R

0 � 1 Sspanned by the functions f � x ! , g � x ! , and h � x ! .

, , ,

1


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra22SymmetricMatrices due 1/22/09 at 2:00 AM


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra23QuadraticForms due 1/23/09 at 2:00 AM

1.(1 pt) setLinearAlgebra23QuadraticForms/ur la 23 1.pgWrite the matrix of the quadratic formQ � x !6�� 6x2

1 � 4x22 � 4x2

3 � 8x1x2 � 1x1x3 � 3x2x3.

A � ��.

2.(1 pt) setLinearAlgebra23QuadraticForms/ur la 23 4.pg

If A � �3 � 1� 1 6 � and Q � x !6� x

bAx,

Then Q � e1 !6� and Q � e2 !$� .

3.(1 pt) setLinearAlgebra23QuadraticForms/ur la 23 5.pg

If A � 7 4 94 � 5 29 2 1

��and Q � x !-� x

bAx,

Then Q � x1 � x2 � x3 !6� x21 � x2

2 � x23 � x1x2 �

x1x3 � x2x3.

4.(1 pt) setLinearAlgebra23QuadraticForms/ur la 23 2.pgFind the eigenvalues of the matrix

A � �6 ' 5 � 0 ' 5� 0 ' 5 6 ' 5 � .

The smaller eigenvalue is λ1 � ,

and the bigger eigenvalue is λ2 � .Classify the quadratic form Q � x !6� xT Ax :� A. Q � x ! is negative definite� B. Q � x ! is positive semidefinite� C. Q � x ! is indefinite� D. Q � x ! is negative semidefinite� E. Q � x ! is positive definite

5.(1 pt) setLinearAlgebra23QuadraticForms/ur la 23 3.pgThe matrix

A � 2 � 2 0� 2 2 00 0 7

��has three distinct eigenvalues, λ1 [ λ2 [ λ3,λ1 � ,λ2 � ,λ3 � .Classify the quadratic form Q � x !6� xT Ax :� A. Q � x ! is negative semidefinite� B. Q � x ! is positive semidefinite� C. Q � x ! is positive definite� D. Q � x ! is negative definite� E. Q � x ! is indefinite


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ARNOLD PIZER rochester problib from CVS Summer 2003Rochester WeBWorK Problem Library WeBWorK assignment LinearAlgebra24SingularValues due 1/24/09 at 2:00 AM


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ARNOLD PIZER rochester problib from CVS Summer 2003doyle/docs/cs.old/raw/ww/pdf-summaries… · ARNOLD PIZER rochester problib from CVS Summer ... determine whether it has a unique