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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Linear Algebra and Its Applications, Exercise 3.1.19

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Linear Algebra and Its Applications, Exercise 3.1.19

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