Linear Algebra and Its Applications, Exercise 2.6.20

Exercise 2.6.20. A nonlinear transformation $f$ from a vector space $V$ to a vector space $W$ is invertible a) if for any $b$ in $W$ there exists some $x$ in $V$ such that $f(x) = b$ and b) if $x$ and $y$ are in $V$ then $f(x) = f(y)$ implies that $x = y$ . Describe which of the following transformations from $\mathbb{R}$ to $\mathbb{R}$ are invertible, and explain why or why not:

a) $f(x) = x^3$

b) $f(x) = e^x$

c) $f(x) = x+11$

d) $f(x) = \cos x$

Answer: a) If $b$ is any real number then $x = \sqrt[3]{b}$ exists and is a unique solution to $x^3 = b$ . The transformation $f(x) = x^3$ is therefore invertible.

b) We have $e^x > 0$ for all $x$ in $\mathbb{R}$ , so if $b \le 0$ then there is no $x$ for which $e^x = b$ . The transformation $f(x) = e^x$ is therefore not invertible.

c) If $b$ is any real number then $b-11$ exists and is a unique solution to $x+11 = b$ . The transformation $f(x) = x+11$ is therefore invertible.

d) We have $-1 \le \cos x \le 1$ for all $x$ in $\mathbb{R}$ so if $b < -1$ or $b > 1$ then there is no $x$ for which $\cos x = b$ . Also note even for $-1 \le b \le 1$ solutions to $\cos x = b$ are not unique, since (for example) $\cos 0 = \cos 2\pi = 1$ . The transformation $f(x) = \cos x$ is therefore not invertible.

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 2.6.20

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Linear Algebra and Its Applications, Exercise 2.6.20

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