AI Tutor
Back to Orthogonal complements and the matrix transpose

Exercise 3

Avg difficulty 2.5

Suppose that WW is the subspace of R4\real^4 defined as the solution set of the equation x1−3x2+5x3−2x4=0.x_1 - 3x_2 + 5x_3 - 2x_4 = 0.

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

  2. Find a basis for WW.

  3. Find a basis for W⊥W^\perp.

  4. In general, how can you easily find a basis for W⊥W^\perp when WW is defined by Ax1+Bx2+Cx3+Dx4=0?Ax_1+Bx_2+Cx_3+Dx_4 = 0?

Austin, Understanding Linear Algebra, §6.2 Orthogonal complements and the matrix transpose, Exercise 3

AI Tutor

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