Linear Algebra Done Right · Problem 3D.21

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · systems-of-linear-equations

Problem

Suppose nn is a positive integer and Aj,k∈FA_{j,k} \in \mathbf{F} for all j,k=1,…,nj, k = 1, \ldots, n. Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables). (a) The trivial solution x1=⋯=xn=0x_1 = \cdots = x_n = 0 is the only solution to the homogeneous system of equations ∑k=1nA1,kxk=0⋮∑k=1nAn,kxk=0.\sum_{k=1}^{n} A_{1,k} x_k = 0 \quad \vdots \quad \sum_{k=1}^{n} A_{n,k} x_k = 0. (b) For every c1,…,cn∈Fc_1, \ldots, c_n \in \mathbf{F}, there exists a solution to the system of equations ∑k=1nA1,kxk=c1⋮∑k=1nAn,kxk=cn.\sum_{k=1}^{n} A_{1,k} x_k = c_1 \quad \vdots \quad \sum_{k=1}^{n} A_{n,k} x_k = c_n.

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