Kunze Linear Algebra · Problem 2.6.2

Vector Spaces

Mathematics — Methods · subspace-membership-linear-combinations

Problem

Let α1=(1,1,−2,1),α2=(3,0,4,−1),α3=(−1,2,5,2).\alpha_1 = (1, 1, -2, 1), \quad \alpha_2 = (3, 0, 4, -1), \quad \alpha_3 = (-1, 2, 5, 2). Let α=(4,−5,9,−7),β=(3,1,−4,4),γ=(−1,1,0,1).\alpha = (4, -5, 9, -7), \quad \beta = (3, 1, -4, 4), \quad \gamma = (-1, 1, 0, 1). (a) Which of the vectors α,β,γ\alpha, \beta, \gamma are in the subspace of R4R^4 spanned by the αi\alpha_i? (b) Which of the vectors α,β,γ\alpha, \beta, \gamma are in the subspace of C4C^4 spanned by the αi\alpha_i? (c) Does this suggest a theorem?

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