LINEAR ALGEBRA
BASIC CONCEPTS
• A Matrix is a rectangular arrays of numbers (real
or complex) enclosed in brackets for instance
a11
a21
.
.
am1
a12
a22
….
….
a1n
a2n
am2
….
amn
• The numbers a11,…amn are called elements.
• Horizontal lines are called rows or row vectors.
• Vertical lines are called columns or column
vectors.
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MATRIX – BASIC CONCEPTS
• A matrix with ‘m’ rows and ‘n’ columns is called
m x n matrix (read m by n matrix).
• In a double script notation of matrix element, ajk,
the first subscript denotes row and second
subscript denoted column.
• Row Matrix has only one row
– [a1, a2,….an]
• Column matrix has only one column.
• Square matrix has same no of rows and columns.
• Diagonal of square matrix is called principal
diagonal.
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MATRIX MULTIPLICATION
Let A =[ajk] be an mxn matrix and B=[bjk] and rxp matrix.
Then the product AB (in this order) is defined only when r=n
and is the mxp matrix C=[cjk]whose elements are
n
cjk=aj1b1k + aj2b2k + …..+ ajnbnk=
a b
ji
ik
i 1
We see that cjk is the inner product (dot product) of the j-th
row vector of the first matrix A, and the k-th column vector
of the secodn matrix B.
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MATRIX MULTIPLICATION
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MATRIX MULTIPLICATION
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SOME EXAMPLES
A=
C=A x B=
1
2
-10
10
4
-2
27
-6
B=
c11 =
c12 =
c21 =
c22 =
2
-3
a11xb11 +
a11xb12 +
a21xb12 +
a21xb12 +
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6
a12xb21
a12xb22
a22xb21
a22xb22
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SYSTEM OF LINEAR EQUATION
A system of m linear equations (or set of m simulatenous
equations) in n unknowns x1, x2,…xn is a set of equations of
the form
a11x1+a12x2+….+a1nxn = b1
a21x1+a22x2+….+a2nxn = b2
.
.
am1x1+am2x2+..+amnxn = bm
The ajk are given numbers, which are called coefficients
of the system. The bi are also given numbers.
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LINEAR EQUATION
Ax = b – Single vector equation where the
coefficient matrix A =[ajk] is the m x n matrix.
A
a11
a21
.
.
am1
a12
a22
….
….
a1n
a2n
am2
….
amn
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x
b
x1
x2
.
.
xn
b1
b2
.
.
bm
10
AUGMENTED MATRIX
The following is the augmented matrix obtained by adding
the b column vector to the A matrix.
a11
a21
.
.
am1
a12
a22
….
….
a1n
a2n
am2
….
amn
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b1
b2
.
.
bm
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GAUSS ELIMINATION
• First step – Elimination of x1 from the second, third and
mth equation. The first equation is called the pivotal
equation in the first step. Using it we eliminate x1 from
second, third and mth equation. This is done by
substracting
– a21/a11 times the first equation from the second equation.
– a31/a11 times the first equation from the third equation etc
• This gives a new set of equations
– a11x1+a12x2+…+a1mxn = b1
–
c22x2+….+c2nxn = b2
–
cm2x2+…cmnxn = bm
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GAUSS ELIMINATION
• The second step is to eliminate x2 from the third…mth
equation in a similar fashion as we did in first step.
• In the third step we eliminate x3 and so on.
• We then have a system of the form
– a11x1+a12x2+…+a1mxn = b1
–
c22x2+….+c2nxn = b2
–
krrxr+…+krnxn = br
–
0 = b r+1
–
.
–
0 = bm
– where r m
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GAUSS ELIMINATION
• We have three possible cases
– No solution if r < m and one of the numbers b r+1,…bm is not
zero.
– Precisely one solution if r=n and b r+1 …bm, if present are zero.
– Infinitely many solution if r < n
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Example
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