Problems of Vector Spaces

7/25/2019 Problems of Vector Spaces

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LINEAR ALGEBRA

Exercise Sheet 03: Vector Spaces

Exercise 1. Show that the set P3 of polynomials a0 + a1x + a2x2 + a3x3 of degree threeor less is a vector space (where a0, a1, a2a3 are any real numbers and x is a variable).Show that the set P(3) of polynomials, a0 + a1x + a2x2 + a3x3, a = 0, (i.e., of degreeexactly three) is not a vector space.

Note that in general the set of polynomials Pn of degree equal to or less than n, n ∈ N is avector space.

Exercise 2. For the vector space R3 decide which of the following are subspaces:

a) {(x ,0,z ) : x, z ∈ R}

b) {(x ,y,z ) : x = 2y, x,y,z ∈ R}

c) {(x ,y,z ) : x = 2y + 5, x,y,z ∈ R}

Exercise 3. Show that the setC3

of all ordered 3-tuples of complex numbers, with theordinary algebraic operations of addition and multiplication by complex numbers, isa vector space.

Exercise 4. Consider the three matrices in the vector space R2×2 given by

A =

1 0

0 0

, B =

1 0

0 1

, C =

1 0

1 1

Show that the set of linear combinations of A, B, C is a subspace of R2×2.

Exercise 5. Check whether or not the vector (3,4,4) is in the span of the set of vectors {(1,2,3), (−1,0,2)} and, if it is, then find the linear combination.

Exercise 6. Prove that the set of vectors {(1,0,0), (0,1,0), (0,0,1), (1,2,3)} is linearlydependent. Prove that the set of vectors {(0,1,0), (0,0,1), (1,2,3)} is linearly indepen-dent.

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Exercise 7. Decide whether the three polynomials p1(x) = 1 − x + x2; p2(x) = 2 +

x; p3(x) = 1 + 2x − x2 are linearly dependent or independent. If they are linearly

dependent then show one linear dependence.

Exercise 8. Prove that the set B = {(1,1,0), (1,2,1), (2,1,0)} is a basis for R3. Find thecoordinates of vector v = (3, −2, 1) with respect to B.

Exercise 9. Find a basis and the dimension for each of the sets below that are subspacesof R4 :

a) {( y, 2y, y, 0) : y ∈ R}

b) {( y, 2y, y, z ) : y, z ∈R

}

c) {( y, 2y, z, 1) : y, z ∈ R}

d) {( y, y + z,3y − 2z,z ) : y, z ∈ R}

Exercise 10. Given vectors u = ( u 1, u 2), v = ( v1, v2) ∈ R2 decide whether the follo-wing operations are inner products:

a) u · v = −2u 1 v1 + 3u 2 v2

b) u · v = 2u 1 v1 + 3u 2 v2

c) u · v = u 1 v1 + u 2

For each operation that is an inner product:

1) Find the length of vectors (1, 0) and (2, −1).

2) Find their inner product.

3) Find the angle between them.

Exercise 11. Consider the operation defined in the vector space of matrices of dimen-sion 2 × 2 with real entries, R2×2, by:

A · B =

a11 a12

a21 a22

·

b11 b12b21 b22

= a11b11 + a12b12 + a21b21 + a22b22

a) Prove that this operation is an inner product in R2×2.

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b) Prove that the matrices A =

1 3

2 − 4

and B =

0 2

3 3

are orthogonal.

c) Consider the right-angled triangle with sides A, B and hypotenuse H. Find Hand confirm that the three sides verify the Pythagorean theorem.

d) Find the angles between A and H and B and H. Do the angles of this triangleadd up to π ?

e) Find the orthogonal projection of matrix B and C =

1 3

2 4

on matrix A.

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Problems of Vector Spaces