Question

Prob 6. Suppose V is a nonzero finite-dimensional vector space and W is infinite-dimensional. Prove that L(V. W) is infinite-dimensional.

This is an Advanced Linear Algebra Question.

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Answer 1

Let (w_k) be a subspace of vectors in W such that {w_1, ellipsis w_n} is linearly independent for each m greaterthanorequalto 1 let {v_1, v_2 ellipsis v_n} be basis for V Define L_K: v rightarrow W by L_k (v_i) = 0 (i = 2, ellipsis n) Now we can easily prove that for each m, {L_1, L_2 ellipsis L_m} is linearly independent Indeed if a_1 L_1 = ellipsis a_m L_m = 0 then a_1 L_1 (v_1) + ellipsis + a_m L_m (v_1) = 0 That is a_1 w_1 + a_2 w_2 + ellipsis a_m w_m = 0 and therefore a_1 = a_2 = ellipsis a_n = 0 therefore L (V, M) is infinite dimension