## 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 $x = \sqrt{b}$ 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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