Unformatted text preview: University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23 Winter 2008
Lecture 8
REVIEW of lecture 7 A system of m linear equations in n unknowns is:
a11 x1
a21 x1
.
.
. + a12 x2
+ a22 x2 + · · · + a1 n x n
+ · · · + a2 n x n = b1
= b2 am1 x1 + am2 x2 + · · · + amn xn = bm
is equivalent to Ax = b where: A= a11
a21
.
.
. a12
a22 am1 am2 · · · a1 n
· · · a2 n , · · · amn x= x1
x2
.
.
. , and b = bm xn and A is called the coeﬃcient matrix,
The system also corresponds to the augmented matrix: [Ab] = · · · a1 n
· · · a2 n b1
b2
.
.
. am1 am2 · · · amn bm a11
a21
.
.
. a12
a22 1 b1
b2
.
.
. . Ax = b corresponds to: a11
a21 a21 a22 + . x1 . x2
.
.
.
.
a1 m
a2 m
↑
↑
column 1 of A
column 2 of A + ··· an1
an2
.
.
. xn anm
↑
column n of A = b1
b2
.
.
.
bm therefore Ax = b has a solutions if and only if b is in the span of
the columns of A.
definition 0.1. (a) Ax = b is consistent if it has at least one solution,
otherwise it is called inconsistent.
(b)
If [H c] is the result of performing successive elementary row operations to [Ab] then we say that [Ab] and [H c] are row equivalents and
we denote it by [Ab] ∼ [H c]
theorem 0.2. a)If [Ab] ∼ [H c] then:
(i) Ax = b has a solution ⇐⇒ H x = c has a solution. (ii) If the solution set of Ax = b exists, then it is the same as the solution
set of H x = c.
b) If [Ab] ∼ [H c] and [H c] is in REF then:
(i) H x = c is inconsistent (has no solution) ⇐⇒ [H c] has a row of
zeros before the partition and a nonzero entry after the partition.
(ii) If H x = c is consistent and H has a pivot in every column ⇒ the
solution is unique.
(iii) If H x = c is consistent and H has columns without pivots ⇒ the
solution set is inﬁnite with as many ’free variables ’ (or parameters) as
non pivotal columns.
A REF of a matrix is not unique, where its RREF is.
2 definition 0.3.
An Elementary Matrix. An elementary matrix is the
result of performing exactly one elementary row operation to an identity
matrix.
example 0.4. example 0.5. Through examples, guess the relationship between the matrix B , obtained by performing one elementary row to a matrix A, and E and
A, where E is the the elementary matrix that corresponds to that elementary
row operation performed on I . 3 011
example 0.6. Reduce the matrix A = 2 2 1 to a RREF H , using
133
only one elementary row operation at a time. Find H in terms of A and the
elementary matrices corresponding to the elementary row operations performed. 4 ...
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 Spring '10
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 Linear Algebra, Algebra, Linear Equations, Equations, Invertible matrix, Elementary matrix, Computer & Mathematical Sciences, elementary Row

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