Prove or disprove: If the columns of B(n×p) ? R are linearly independent as well as those of A, then so are the columns of AB (for A(m×n) ? R ).
Let B = [b1,b2,…,bp], where bi (1 ? i ? p) is the ith column vector of B. Further, AB = [Ab1,Ab2,…,Abp] . Let us assume that the columns of a mx n matrix A and a nxp matrix B with real entries are linearly independent while the columns of AB are linearly dependent. Then there exist real scalars xi (1 ? i ? p), not all zero such that x1Ab1+x2 Ab2 +…+xp Abp = 0. Then A(x1b1 +x2b2+…+xpbp) = 0 so that x1b1 +x2b2+…+xpbp = 0. However, since the columns of B are linearly independent, this means that all the xi s are 0 which is a contradiction. Hence the columns of AB are also linearly independent.
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