Multivariable calculus: Exercises

07 Functions of multiple variables, part 3

Exercise 1: Matrix/vector representations of linear functions

Consider the linear function \(f: \mathbb{R}^2 \to \mathbb{R}^2\) given by

\[ f(x,y) = (2x - y + 4, x + 3y - 1). \]

As I mentioned in class, this is called a linear function because it only contains terms of degrees \(0\) and \(1\).

If we blur the distinction between a point \((x,y)\) and its position vector

\[ \begin{bmatrix} x \\ y \end{bmatrix}, \]

then we can rewrite the formula for \(f\) as

\[ f\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2 & -1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 4 \\ -1 \end{bmatrix}. \]

Check for yourself that if you multiply out the right-hand side, you will get

\[ f\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix}2x - y + 4 \\ x + 3y - 1 \end{bmatrix}, \]

which is the same as the original formula for \(f\), but written in a different way.

The same procedure works in higher dimensions, as you will discover in the following exercises:

In each of the following, convert the given formula for a linear function into a matrix/vector representation as shown above.

  1. \(f: \mathbb{R}^2 \to \mathbb{R}^2\), \(f(x,y) = (x + 2y - 3, 4x - y + 5)\)
  2. \(g: \mathbb{R}^2 \to \mathbb{R}^2\), \(g(x,y) = (3x - 2y , -x + 5y - 4)\)
  3. \(h: \mathbb{R}^3 \to \mathbb{R}^3\), \(h(x, y, z) = (x + 2y - z + 1, 3x - y + 4z - 2, -2x + 5y + z + 3)\)
  4. \(k: \mathbb{R}^2 \to \mathbb{R}^3\), \(k(x, y) = (x - 4y + 2, -2x + 3y - 1, 3x - y)\)

Now go backwards. For each of the following, convert the given matrix/vector representation into a formula for a linear function.

  1. \(r: \mathbb{R}^2 \to \mathbb{R}^2\), \(r\left( \begin{bmatrix}x \\ y \end{bmatrix} \right) = \begin{bmatrix}1 & 4 \\ -2 & 3 \end{bmatrix} \begin{bmatrix}x \\ y \end{bmatrix} + \begin{bmatrix} -3 \\ 2 \end{bmatrix}\)

  2. \(s: \mathbb{R}^2 \to \mathbb{R}^2\), \(s\left( \begin{bmatrix}x \\ y \end{bmatrix} \right) = \begin{bmatrix}0 & 5 \\ -1 & 2 \end{bmatrix} \begin{bmatrix}x \\ y \end{bmatrix} + \begin{bmatrix} 1 \\ -4 \end{bmatrix}\)

  3. \(t: \mathbb{R}^3 \to \mathbb{R}^3\), \(t\left( \begin{bmatrix}x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix}2 & -1 & 3 \\ -4 & 5 & 0 \\ 1 & 2 & -2 \end{bmatrix} \begin{bmatrix}x \\ y \\ z \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}\)

  4. \(u: \mathbb{R}^2 \to \mathbb{R}^3\), \(u\left( \begin{bmatrix}x \\ y\end{bmatrix} \right) = \begin{bmatrix}1 & -2 \\ 3 & 0 \\ -1 & 4 \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} + \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}\)

  1. \(f\left( \begin{bmatrix}x \\ y\end{bmatrix} \right) = \begin{bmatrix}1 & 2 \\ 4 & -1 \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} + \begin{bmatrix} -3 \\ 5 \end{bmatrix}\)

  2. \(g\left( \begin{bmatrix}x \\ y\end{bmatrix} \right) = \begin{bmatrix}3 & -2 \\ -1 & 5 \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} + \begin{bmatrix} 0 \\ -4 \end{bmatrix}\)

  3. \(h\left( \begin{bmatrix}x \\ y \\ z\end{bmatrix} \right) = \begin{bmatrix}1 & 2 & -1 \\ 3 & -1 & 4 \\ -2 & 5 & 1 \end{bmatrix} \begin{bmatrix}x \\ y \\ z\end{bmatrix} + \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix}\)

  4. \(k\left( \begin{bmatrix}x \\ y\end{bmatrix} \right) = \begin{bmatrix}1 & -4 \\ -2 & 3 \\ 3 & -1 \end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} + \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}\)

  5. \(r(x,y) = (x + 4y - 3, -2x + 3y + 2)\)

  6. \(s(x,y) = (5y + 1, -x + 2y - 4)\)

  7. \(t(x,y,z) = (2x - y + 3z, -4x + 5y +1, x + 2y - 2z - 1)\)

  8. \(u(x,y) = (x - 2y + 2, 3x - 1, -x + 4y + 3)\)

Exercise 2: Functions \(\mathbb{R} \to \mathbb{R}\) as transformations

The following exercise is meant to give you practice with the ideas in Exercise 1 of the slides.

Describe the action of the following functions \(\mathbb{R} \to \mathbb{R}\) as physical transformations of the real line into itself. Draw pictures!

  1. \(f(x) = -x^2\)
  2. \(g(x) = -2x + 2\)
  3. \(h(x) = e^x\)

These are nearly impossible to plot on a computer. You’ll have to catch me after class and show me your sketches to check your work.

Exercise 3: Functions \(\mathbb{R} \to \mathbb{R}^2\) as transformations

The following exercise is meant to give you practice with the ideas in Exercise 2 of the slides.

Describe the action of the following functions \(\mathbb{R} \to \mathbb{R}^2\) as physical transformations of the real line into the plane. Draw pictures!

  1. \(f(t) = (-t^2, t)\)
  2. \(g(t) = (-2t + 2, 3t)\)
  3. \(h(t) = (e^t, t^2)\)
  4. \(k(t) = (\cos{t}, 2\sin{t})\)

Exercise 4: Functions \(\mathbb{R}^2 \to \mathbb{R}^2\) as transformations

The following exercise is meant to give you practice with the ideas in Exercises 4-6 of the slides.

Describe the action of the following functions \(\mathbb{R}^2 \to \mathbb{R}^2\) as physical transformations of the plane into itself. Draw pictures!

  1. \(f(x, y) = (-x + 1, x+y)\)
  2. \(g(x, y) = (x, e^x + y)\)
  3. \(h(x,y) = (xy, x)\)