Linear Algebra and Its Applications, Exercise 3.1.8

Exercise 3.1.8. Suppose that $V$ and $W$ are orthogonal subspaces. Show that their intersection consists only of the zero vector.

Answer: If $V$ and $W$ are orthogonal then we have $v^Tw = 0$ for any vectors $v$ in $V$ and $w$ in $W$ . Suppose that $x$ is an element of both $V$ and $W$ . Then we have $x^Tx = 0$ . But this means that $\|x\| = 0$ and thus that $x = 0$ .

So if $V$ and $W$ are orthogonal then $V \cap W = \{0\}$ .

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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