Let v1, v2,..., vm be a set of linearly independent vectors in a vector space V and

Chapter 4, Problem 47

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Let v1, v2,..., vm be a set of linearly independent vectors in a vector space V and suppose that the vectors u1, u2,..., un are each linear combinations of them. It follows that we can write uk = m i=1 aikvi, k = 1, 2,..., n, for appropriate constants aik . (a) If n > m, prove that {u1, u2,..., un} is linearlydependent on V.(b) If n = m, prove that {u1, u2,..., un} is linearlyindependent in V if and only if det[ai j] = 0.(c) If n < m, prove that {u1, u2,..., un} is linearlyindependent in V if and only if rank(A) = n,where A = [ai j].(d) Which result from this section do these resultsgeneralize?