Math 260, Fall 2019, 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

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

1. $S_{\mathcal{B}\rightarrow\mathcal{C}}$ (the change-of-basis matrix transforming $\mathcal{B}$-coordinates into $\mathcal{C}$-coordinates).
2. $\delta_{i,j}$ (the Kronecker delta).
3. Orthonormal set of vectors.

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

1. Theorem concerning the linearity of coordinates.
2. Theorem relating $S_{\mathcal{B}\rightarrow\mathcal{C}}$, $[\vec{v}]_{\mathcal{B}}$, and $[\vec{v}]_{\mathcal{C}}$ (a.k.a. the transformation law for vectors).

Carefully describe the following algorithms:

1. Algorithm to compute $[\vec{v}]_{\mathcal{B}}$ (given a basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$ and a vector $\vec{v}$ lying in the span of $\mathcal{B}$).
2. Algorithm to compute $S_{\mathcal{B}\rightarrow\mathcal{C}}$ (given two bases $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$ and $\mathcal{C}=(\vec{w}_1,\dots,\vec{w}_k)$ for the same space).

Solve the following problems:

1. Section 3.4, problems 5, 7, 9, 11, 15, and 17.
2. Section 5.1, problem 16.
3. Working in $\mathbb{R}^5$, consider the bases $$\mathcal{B}=\left(\begin{bmatrix}1\\0\\1\\2\\3\end{bmatrix},\begin{bmatrix}1\\1\\0\\2\\1\end{bmatrix}\right)\quad\text{and}\quad\mathcal{C}=\left(\begin{bmatrix}3\\1\\2\\6\\7\end{bmatrix},\begin{bmatrix}5\\2\\3\\10\\11\end{bmatrix}\right).$$

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

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

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

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

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