Math 260, Fall 2018, Assignment 10

The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.

- Saint Augustine

Read:

1. Section 3.4.

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

1. Coordinates (of a vector $\vec{v}$ with respect to a basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$).
2. $\left[\vec{v}\right]_{\mathcal{B}}$ (also known as the coordinate vector of $\vec{v}$ with respect to the basis $\mathcal{B}$).
3. $S_{\mathcal{A}\rightarrow\mathcal{B}}$ (also known as the change-of-basis or change-of-coordinate matrix from basis $\mathcal{A}$ to basis $\mathcal{B}$.

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

1. Theorem concerning uniqueness of coordinates.
2. Theorem concerning linearity of the coordinate map.
3. Theorem concerning linearity of the inverse coordinate map.
4. Theorem concerning linearity of coordinate transformations.

Carefully describe the following algorithms:

1. Algorithm to compute the coordinates of a vector $\vec{v}$ with respect to a basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$.
2. Algorithm to compute the change-of-basis matrix $S_{\mathcal{A}\rightarrow\mathcal{B}}$.

Solve the following problems:

1. Section 3.4, problems 5, 7, 9, 11, 15, and 17.
2. Working in $\mathbb{R}^5$, consider the bases $$\mathcal{A}=\left(\begin{bmatrix}1\\0\\1\\2\\3\end{bmatrix},\begin{bmatrix}1\\1\\0\\2\\1\end{bmatrix}\right)\quad\text{and}\quad\mathcal{B}=\left(\begin{bmatrix}3\\1\\2\\6\\7\end{bmatrix},\begin{bmatrix}5\\2\\3\\10\\11\end{bmatrix}\right).$$

(i) Verify that $\mathcal{A}$ and $\mathcal{B}$ span the same plane.

(ii) Compute the change-of-basis matrices $S_{\mathcal{A}\rightarrow\mathcal{B}}$ and $S_{\mathcal{B}\rightarrow\mathcal{A}}$.

(iii) Compute the products $S_{\mathcal{A}\rightarrow\mathcal{B}}S_{\mathcal{B}\rightarrow\mathcal{A}}$ and $S_{\mathcal{B}\rightarrow\mathcal{A}}S_{\mathcal{A}\rightarrow\mathcal{B}}$. Do the results surprise you?

(iv) Consider the vectors $\vec{u}_1$ and $\vec{u}_2$ whose $\mathcal{A}$-coordinate vectors are $$\left[\vec{u}_1\right]_{\mathcal{A}}=\begin{bmatrix}3\\-1\end{bmatrix}\quad\text{and}\quad\left[\vec{u}_2\right]_{\mathcal{A}}=\begin{bmatrix}1\\2\end{bmatrix}.$$ Compute the $\mathcal{B}$-coordinate vectors of $\vec{u}_1$ and $\vec{u}_2$.

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