## Linear Algebra and Its Applications, Exercise 3.1.19

Exercise 3.1.19. State whether each of the following is true or false:

(a) If the subspaces $V$ and $W$ are orthogonal, then $V^\perp$ and $W^\perp$ are also orthogonal.

(b) If $V$ is orthogonal to $W$ and $W$ orthogonal to $Z$ then $V$ is orthogonal to $Z$.

Answer: (a) In $\mathbb{R}^n$ suppose that $V = W = \{0\}$. Then since the zero vector is orthogonal to itself $V$ is orthogonal to $W = V$. However we have $V^\perp = W^\perp = \mathbb{R}^n$ and $\mathbb{R}^n$ is not orthogonal to itself, so $V^\perp$ and $W^\perp$ are not orthogonal. The statement is false.

(b) Suppose that $V = Z = \mathbb{R}^n$ and $W = \{0\}$. Then $V$ is orthogonal to $W$ and $W$ is orthogonal to $Z$ but $V$ is not orthogonal to $Z$ because $\mathbb{R}^n$ is not orthogonal to itself. The statement is false.

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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