Math 260, Fall 2019, Assignment 10
From cartan.math.umb.edu
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:[edit]
- $S_{\mathcal{B}\rightarrow\mathcal{C}}$ (the change-of-basis matrix transforming $\mathcal{B}$-coordinates into $\mathcal{C}$-coordinates).
- $\delta_{i,j}$ (the Kronecker delta).
- Orthonormal set of vectors.
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning the linearity of coordinates.
- 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:[edit]
- 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 $\mathcal{B}$).
- 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:[edit]
- Section 3.4, problems 5, 7, 9, 11, 15, and 17.
- Section 5.1, problem 16.
- 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 $\vec{u}_2$.
