Kunze Linear Algebra · Problem 2.4.4

Vector Spaces

Mathematics — Methods · change-of-basis

Problem

Let WW be the subspace of C3C^3 spanned by α1=(1,0,i)\alpha_1 = (1, 0, i) and α2=(1+i,1,−1)\alpha_2 = (1+i, 1, -1). (a) Show that α1\alpha_1 and α2\alpha_2 form a basis for WW. (b) Show that the vectors β1=(1,1,0)\beta_1 = (1, 1, 0) and β2=(1,i,1+i)\beta_2 = (1, i, 1+i) are in WW and form another basis for WW. (c) What are the coordinates of α1\alpha_1 and α2\alpha_2 in the ordered basis {β1,β2}\{\beta_1, \beta_2\} for WW?

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