## 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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