Linear Algebra and Its Applications · Problem 3.4.32

Orthogonality — Gilbert Strang

Mathematics · orthogonal-complement-subspace

Problem

(a) Find a basis for the subspace S\mathbf{S} in R4\mathbf{R}^4 spanned by all solutions of x1+x2+x3−x4=0.x_1 + x_2 + x_3 - x_4 = 0. (b) Find a basis for the orthogonal complement S⊥\mathbf{S}^\perp. (c) Find b1b_1 in S\mathbf{S} and b2b_2 in S⊥\mathbf{S}^\perp so that b1+b2=b=(1,1,1,1)b_1 + b_2 = b = (1, 1, 1, 1).

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