Page 76 - Linear Algebra Done Right
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Chapter 3. Linear Maps
62
Suppose n is a positive integer and a i,j ∈ F for i, j = 1,...,n.
26.
Prove that the following are equivalent:
(a) The trivial solution x 1 = ··· = x n = 0 is the only solution
to the homogeneous system of equations
n
a 1,k x k = 0
k=1
. . .
n
a n,k x k = 0.
k=1
(b) For every c 1 ,...,c n ∈ F, there exists a solution to the sys-
tem of equations
n
a 1,k x k = c 1
k=1
. . .
n
a n,k x k = c n .
k=1
Note that here we have the same number of equations as vari-
ables.
