Math 260, Fall 2019, Assignment 5

I tell them that if they will occupy themselves with the study of mathematics, they will find in it the best remedy against the lusts of the flesh.

- Thomas Mann, The Magic Mountain

Read:

1. Section 2.1.
2. Section 2.2.

Carefully define the following terms, then give one example and one non-example of each:

1. Transformation (from $\mathbb{R}^m$ to $\mathbb{R}^n$).
2. Linear (transformation).

Carefully state the following theorems (you do not need to prove them):

1. Description of matrix multiplication in terms of columns (i.e. $\begin{bmatrix}\vec{v}_1&\vec{v}_2&\dots&\vec{v}_m\end{bmatrix}\begin{bmatrix}x_1\\x_2\\\dots\\x_m\end{bmatrix}=\dots$).
2. Theorem relating linearity to representation by matrix multiplication.
3. Formula for the matrix of a linear transformation (in terms of the values $f(\vec{e}_1),\dots,f(\vec{e}_m)$).
4. Formula for the matrix representing a rotation in $\mathbb{R}^2$.
5. Theorem concerning orthogonal decomposition of a vector $\vec{x}$ with respect to another, non-zero, vector $\vec{v}$.
6. Formulas for the parallel and perpendicular components of $\vec{x}$ with respect to $\vec{v}$.
7. Formula for the matrix representing $\mathrm{Proj}_{\vec{v}}$.

Solve the following problems:

1. Section 2.1, problems 1, 3, 5, 25, 27, 29, and 42.
2. Section 2.2, problems 10, 19, and 21.
3. Define a transformation $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ by the formula $f(\vec{x})=\vec{x}$. (This is known as the identity transformation.) Is $f$ linear? If so, what is its matrix?
4. Repeat the previous problem with $\mathbb{R}^3$ in place of $\mathbb{R}^2$. Then repeat with $\mathbb{R}^n$ in place of $\mathbb{R}^2$.

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