Math 260, Fall 2018, 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 2.3.

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

1. $\mathrm{Proj}_{\vec v}(\vec x)$.
2. Composition (of two functions $f:\mathbb{R}^n\rightarrow\mathbb{R}^p$ and $g:\mathbb{R}^m\rightarrow\mathbb{R}^n$).
3. Inverse (of a function $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$).

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

1. Theorem concerning parallel and perpendicular components.
2. Formula for the matrix representing a rotation in $\mathbb{R}^2$.
3. Theorem concerning linearity of the composition of two linear transformations.
4. Formula for the matrix representing the composition of two linear transformations.

Solve the following problems:

1. Section 2.2, problems 6, and 10.
2. Section 2.3, problems 33, 35, 39, 43, 45, and 46.

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Solutions:

Definitions:

1. Given a non-zero vector $\vec{v}\in\mathbb{R}^n$, define a linear transformation $\mathrm{Proj}_{\vec v}:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by the formula $\mathrm{Proj}_{\vec v}(\vec x) = \vec{x}^{\parallel} = \frac{\vec{x}\cdot\vec{v}}{\vec{v}\cdot{\vec{v}}}\vec{v}$. We refer to this transformation as orthogonal projection along $\vec{v}$. (Note that in spite of the name, the output of the orthogonal projection is parallel to $\vec v$, not perpendicular to it.)
2. Given a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^p$ and a function $g:\mathbb{R}^m\rightarrow\mathbb{R}^n$, the composition of $f$ and $g$ is the function $(f\circ g):\mathbb{R}^m\rightarrow\mathbb{R}^p$ defined by the formula $(f\circ g)(\vec x)=f(g(\vec x))$.
3. Given a function $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$, an inverse of $f$ is another function $f^{-1}:\mathbb{R}^n\rightarrow\mathbb{R}^m$ such that for any $\vec x$, we have $f^{-1}(f(\vec x))=\vec x$ and for any $\vec y$ we have $f(f^{-1}(\vec y))=\vec y$.

Theorems:

1. Given a non-zero vector $\vec v$ and any other vector $\vec x$, there exist unique vectors $\vec{x}^{\parallel}$ and $\vec{x}^{\perp}$ such that (i) $\vec{x}^{\parallel}$ is a scalar multiple of $\vec{v}$, (ii) $\vec{x}^{\perp}$ is orthogonal to $\vec v$, and (iii) $\vec x = \vec{x}^{\parallel}+\vec{x}^{\perp}$. These are given by the formulas $$\vec{x}^{\parallel}=\frac{\vec{x}\cdot\vec{v}}{\vec{v}\cdot\vec{v}}\vec{v}\qquad\vec{x}^{\perp}=x-\frac{\vec{x}\cdot\vec{v}}{\vec{v}\cdot\vec{v}}\vec{v}.$$
2. The matrix of rotation through the counterclockwise angle $\theta$ in $\mathbb{R}^2$ is $\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{bmatrix}.$
3. The composition of two linear transformations is linear.
4. The matrix representing the composite transformation is the product of the matrices representing the transformations. In symbols, let us denote the matrix represeting any linear transformation $f$ by $[f]$; then $$[f\circ g] = [f][g].$$