Math 260, Spring 2017, Assignment 5

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

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

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

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

1. Theorem concerning representability of linear transformations by matrices.
2. Formula for the matrix representing a linear transformation.
3. Formula for the orthogonal decomposition of a vector with respect to a line.

Solve the following problems:

1. Section 2.1, problems 1, 3, 5, 16, 17, 19, 25, 27, and 29.
2. Section 2.2, problems 5 and 26 (you will need to skim the section for the definitions of "reflection," "scaling," and "shearing," and for the matrices that represent these transformations).

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Solutions:

Definitions:

1. A linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^n$ is a function $T:\mathbb{R}^m\rightarrow\mathbb{R}^m$ such that, for every $\vec{x},\vec{y}\in\mathbb{R}^m$ and every $c\in\mathbb{R}$, (1) $T(\vec{x}+\vec{y}) = T(\vec{x}) + T(\vec{y})$, and (2) $T(c\vec{x}) = cT(\vec{x})$. For example, rotation in the plane is linear, while the transformation $T:\mathbb{R}^1\rightarrow\mathbb{R}^1$ defined by the formula $T\left(\begin{bmatrix}x\end{bmatrix}\right) = \begin{bmatrix}x^2\end{bmatrix}$ is not.
2. Let $T:\mathbb{R}^m\rightarrow\mathbb{R}^n$ be a linear transformation. A matrix $M$ is said to represent $T$ is, for every $\vec{x}\in\mathbb{R}^m$, $T(\vec{x}) = M\vec{x}$. For example, counterclockwise rotation of the plane through $90^{\circ}$ is represented by the matrix $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ but not by any other matrix.

Theorems:

1. Theorem concerning representability of linear transformations by matrices: Every linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^m$ can be represented by a matrix.
2. Formula for the matrix representing a linear transformation: A linear transformation $T:\mathbb{R}^m\rightarrow\mathbb{R}^n$ is represented (in standard coordinates) by one and only one matrix, namely the matrix $\begin{bmatrix}T\end{bmatrix} = \begin{bmatrix}T(\vec{e}_1) | \dots | T(\vec{e}_n)\end{bmatrix}$ whose columns are the values of $T$ on the standard basis vectors.
3. Formula for the orthogonal decomposition of a vector with respect to a line: Let $L$ be a line through the origin in $\mathbb{R}^2$, and let $\vec{x}$ be any vector in $\mathbb{R}^2$. Then there exist unique vectors $\vec{x}^{\parallel}$ and $\vec{x}^{\perp}$ such that $\vec{x}^{\parallel}$ lies along $L$, $\vec{x}^{\perp}$ is perpendicular to $L$, and $\vec{x}=\vec{x}^{\parallel}+\vec{x}^{\perp}$. These are given by the formulas $\vec{x}^{\parallel} = (\vec{x}\cdot\vec{u})\vec{u}$ and $\vec{x}^{\perp} = \vec{x}-\vec{x}^{\parallel}$, where $\vec{u}$ is a unit vector along $L$. (A unit vector can be obtained by first finding any vector along $L$, then dividing this by its length.)