## 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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