**Unformatted text preview: **University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23 Winter 2008
Lecture 4
Sophie Chrysostomou
1.3 Linear Systems and Matrices Linear Systems
definition 0.1. An m × n linear system of equations is a system of m linear
equations in n variables:
a11 x1 + a12 x2 + ... + a1n xn
a21 x1 + a22 x2 + ... + a2n xn
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b2
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. am1 x1 + am2 x2 + ... + amn xn = bm
where x1 , · · · xn are the unknowns and aij are constant real numbers for all
i = 1, 2, · · · , m, and j = 1, 2, · · · n.
example 0.2. 2x1 + 3x2 = 9
x1 − 2x2 = 1 In order to solve such a system we perform in some sequence the following
three operations:
1. Switch two equations.
2. Multiply an equation by a nonzero constant. 1 3. Replace an equation by that equation plus a multiple of another equation. definition 0.3.
1. An m × n matrix is an ordered rectangular array, of real numbers, with
m rows and n columns. a11 a12
···
a1 n a21 a22
···
a2 n .
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am1 am2 · · · amn
2. We may denote the matrix by giving it a name, say A, and write
A = [aij ], where aij is the entry in the ith row and j th column of A. b1 b2 3. An m × 1 matrix . is called a column vector in Rm .
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bm
4. A 1 × n matrix c1 c2 · · · cn is called a row vector in Rn . 5. A matrix is called a square matrix if the number of its rows and the
number of its columns are equal.
2 6. A square matrix A is called a diagonal matrix if
aij = 0 for all i = j
7. A square matrix A is called an identity matrix if
aij = 0
1 for all
for all i=j
i=j 8. A square matrix A is called an upper triangular matrix if
aij = 0 for all i > j
9. A square matrix A is called a lower triangular matrix if
aij = 0 for all i < j 3 ...

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