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Back to Orthogonal complements and the matrix transpose

Exercise 1

Avg difficulty 2.5

Suppose that WW is a subspace of R4\real^4 with basis w1=[−222−4],w2=[−235−5].\wvec_1=\fourvec{-2}22{-4},\hspace{24pt} \wvec_2=\fourvec{-2}35{-5}.

  1. What are the dimensions dim⁡W\dim W and dim⁡W⊥\dim W^\perp?

  2. Find a basis for W⊥W^\perp.

  3. Verify that each of the basis vectors for W⊥W^\perp are orthogonal to w1\wvec_1 and w2\wvec_2.

Austin, Understanding Linear Algebra, §6.2 Orthogonal complements and the matrix transpose, Exercise 1

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.