Linear Algebra and Its Applications, Exercise 3.1.22

Exercise 3.1.22. Consider the equation $x_1+x_2+x_3+x_4 = 0$ and the subspace $S$ of $\mathbb{R}^n$ containing all vectors that satisfy it. Find a basis for $S^\perp$, the orthogonal complement of $S$.

Answer:  $S$ is the nullspace of the linear system $Ax = 0$ where

$A = \begin{bmatrix} 1&1&1&1 \end{bmatrix}$

Since $S$ is the nullspace of $A$ its orthogonal complement $S^\perp$ is the row space of $A$, which is spanned by the vector $(1, 1, 1, 1)$ (the first and only row of $A$). The vector

$\begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}$

is thus a basis for $S^\perp$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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