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MATH298:
MATHEMATICS FOR CIVIL ENGINEERS
Web: http://www.maths.liv.ac.uk/~vadim/M298
Lecturer: Prof. Vadim Biktashev
(pronounced nearly as: big tah, chef!)
Office: 415A, Mathematical Sciences Building
Phone: 794 4006
Email: [email protected]
Lectures:
Mon 1012
Wed 0911
Office hours:
Mon 1314
Thu 1314
Recommended texts for Part A:
K. A. Stroud: Engineering Mathematics (McMil-lan)
K. A. Stroud: Further Engineering Mathematics(McMillan)
E. Kreyszig: Advanced Engineering Mathematics(Wiley, New
York)
Check http://www.livmathsbooks.co.uk/
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MATH298 Set 1 2008/01/21
MATRICES
A system of linear algebraicequations:
2x + 3y = 15x + 4y = 1
Solution: x = 1, y = 1.
Another system of linear al-gebraic equations:
2a + 3b = 15a + 4b = 1
Solution: a = 1, b = 1.
Do we have to solve the second system if we know thesolution to
the first?
Both systems have:
the same matrix of coefficients: A =
2 35 4
and
the same augmented matrix A =
2 3 15 4 1
and
the same vector of solution 11 .
Example (DIY)
System of equations
5x 2y + z = 03x + 4z = 7
has matrices
A =
5 2 13 0 4
and A =
5 2 1 03 0 4 7
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Definition
An mn matrix is a rectangular table of numbers (vari-
ables, expressions . . . ) with m rows and n columns:
A =
a11 a12 . . . a1na21 a22 . . . a2n
. . . . . .am1 am2 . . . amn
= [ajk ] = [a]
A column of matrix A:
a12
a22.
am2
A row of matrix A:
am1 am2 . . . amn
Example: Sales figures
M T W Th F S
40 33 81 0 21 47 I0 12 78 50 50 96 II
10 0 0 27 43 78 III
for products I, II and III during 6 days of a weekis a 3x6
matrix:
A =
40 33 81 0 21 470 12 78 50 50 96
10 0 0 27 43 78
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Shapes and sizes
Square matrices
A square matrix: # of rows = # of columns: 1 2 34 5 6
7 8 9
,
A BC D
,
x
are square matrices of sizes 3 3, 2 2 and 1 1.
A matrix is rectangular if it is not squre.
Vectors
A row vector is a matrix with only one row:
a =
a1 a2 . . . an
e.g. a =
5 3 12
A column vector is a matrix with only one column:
b =
b1b2...
bm
e.g. b = 407
A scalar is a number, i.e. not a matrix neither a vector.But
sometimes a matrix 1 1, like
c =
6
would be called a scalar too.
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Transposition
Vectors: Tr. converts rows to columns and vice versa:
a =
5 3 12
aT =
5312
b =
407
bT = 4 0 7
Matrices: IfA is an m n matrix, then AT is an nmmatrix so that
every i-th column of AT is transposedi-th row ofA:
A = [ajk] =
a11 a12 . . . a1na21 a22 . . . a2n
. . . . . .am1 am2 . . . amn
AT = [ajk ] =
a11 a21 . . . am1a12 a22 . . . am2
. . . . . .a1n a2n . . . amn
so ajk = akj.
Example:
A =
5 8 14 0 0
AT =
5 48 01 0
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Equality
Definition Two matrices are equal if they have the same
size and corresponding elements are equal.Examples
A =
0.1 1 0
3 10 0.5
B =
0.1 1 0
3 10 0.5
C =
0.1 1 0
3 11 0.5
D =
0.1 1 0
3 10 0.5
1 0 0
By definition, A = B, A = C (since c22 = a22), andA = D
(different sizes).
Addition
Definiton If two matrices A and B have the same size,then their
sum A+B is obtained by adding correspond-ing entries. Example:
DIY
A =
4 6 3
0 1 2
, B =
5 1 03 1 0
A + B = 1 5 33 2 2
NB: cant add matrices of different sizes!
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Scalar multiplication of matrices
To multiply a matrix A [ajk ] by a scalar c, multiply
each element by it:cA = [cajk ].
In particular: negative of a matrix is
A (1)A,
negative of a multiple is
(k)A = kA,and difference of two matrices is
A + (1)B AB.
Example: If
A = 2.7 1.8
0 0.99.0 4.5
then A = 2.7 1.8
0 0.99.0 4.5
,
10
9A =
3 20 1
10 5
, and 0A =
0 00 0
0 0
.
Example (DIY):
A = 4 6 3
0 1 2
, then 2A = 8 12 6
0 2 4
.
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Some properties
IfA, B . . . are matrices of the same size,0 is a matrix of the
same size with all elements zeros,and c is c, k are scalars,
then
A + B = B+ A.
A+(B+C) = (A+B)+C,
A+ 0 = A,
AA = 0,
(A + B)T = AT + BT.
c(A+ B) = cA+ cB,
(c + k)A = cA+ kA,
c(kA) = (ck)A = ckA,
1A = A,
(cA)T = cAT,
(AT)T = A
Example: Simplify the matrix expression
A + 2
1
2A
T
T
Solution:
= A + 2
1
2
A
TT
= A+ (1)A = 0.
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