16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.3.2

Linear Transformations

Mathematics — Linear Algebra · isomorphism-subspace-questions

Problem

Let VV be a vector space over the field of complex numbers, and suppose there is an isomorphism TT of VV onto C3C^3. Let α1,α2,α3,α4\alpha_1, \alpha_2, \alpha_3, \alpha_4 be vectors in VV such that Tα1=(1,0,i),Tα2=(−2,1+i,0),T\alpha_1 = (1, 0, i), \quad T\alpha_2 = (-2, 1+i, 0), Tα3=(−1,1,1),Tα4=(2,i,3).T\alpha_3 = (-1, 1, 1), \quad T\alpha_4 = (\sqrt{2}, i, 3). (a) Is α1\alpha_1 in the subspace spanned by α2\alpha_2 and α3\alpha_3? (b) Let W1W_1 be the subspace spanned by α1\alpha_1 and α2\alpha_2, and let W2W_2 be the subspace spanned by α3\alpha_3 and α4\alpha_4. What is the intersection of W1W_1 and W2W_2? (c) Find a basis for the subspace of VV spanned by the four vectors αi\alpha_i.

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