Page 70 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes where a, b, c are arbitrary scalars in F. Finally let W be the subset of V consisting of all matrices
2
of the form
a 0
0 b
where a, b are arbitrary scalars in F. Then W , W are subspaces of V.
1 2

Example 7: The solution space of a system of homogeneous linear equations. Let us
consider the simultaneous equations involving n unknown x ’s.
i
a x a x ... a x 0
11 1 12 2 1n n
a x a x 2 ... a x 0
22
2n n
21 1
. . .
. . .
. . .
. . .
. . .
a x a x ... a x 0
m 1 1 m 2 2 mn n
In matrix form we write the equation as

AX = 0
where A is a m × n matrix over the field F as all a ij A for i = 1 to m and j = 1 to n. Then the set of
all n × 1 matrices X over the field such that
AX = 0
is a subspace of the space of all n × 1 matrices over F. To prove this we must show that
A(ax + y) = 0

when AX = 0 and AY = 0.
and C is an arbitrary scalar in F.
Consider a matrix A an m × n matrix over F and B and C are n × p matrices over F, then

A(a B + C) = a (AB) + AC
for each scalar a in F. Now

A aB C A aB C
ij ik kj
k

aA B A C
ik kj ik kj
k

a A B kj A C kj
ik
ik
k k
a AB AC
ij ij
aAB AC
ij

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