Problem 1. Let A E Fm,n be such that the associated linear map TEL(F, Fm) has rank r. (We are using standard bases.) Prove the following: (a) A has r linearly independent columns. (b) The system of m homogeneous linear equations in n variables n ΣAijj = 0, i=1,2,...,m, j=1 has a non-trivial solution if and only if r < n. (c) The system of equations n Aijdj=bi, i=1,2,...,m, j=1 has a solution for each (b₁,b₂,..., bm) E Fm if and only if r = m.
Problem 1. Let A E Fm,n be such that the associated linear map TEL(F, Fm) has rank r. (We are using standard bases.) Prove the following: (a) A has r linearly independent columns. (b) The system of m homogeneous linear equations in n variables n ΣAijj = 0, i=1,2,...,m, j=1 has a non-trivial solution if and only if r < n. (c) The system of equations n Aijdj=bi, i=1,2,...,m, j=1 has a solution for each (b₁,b₂,..., bm) E Fm if and only if r = m.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In the problems below, F denotes R or C.
Problem 1. Let A € Fm,n be such that the associated linear map T = L(F¹, Fm) has rank
r. (We are using standard bases.) Prove the following:
(a) A has r linearly independent columns.
(b) The system of m homogeneous linear equations in n variables
N
Aijxj = 0, i = 1, 2, . . . , m,
j=1
has a non-trivial solution if and only if r <n.
(c) The system of equations
n
AijXj = bi, i = 1, 2, . . ., m,
j=1
has a solution for each (b₁,b2,..., bm) € Fm if and only if r = m.
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