Math 260, Spring 2017, Assignment 10
From cartan.math.umb.edu
The moving power of mathematical invention is not reasoning but the imagination.
- - Augustus de Morgan
Carefully define the following terms, then give one example and one non-example of each:[edit]
- $\left\lbrack\vec{x}\right\rbrack_{\mathcal{B}}$ (a.k.a. the coordinate vector of $\vec{x}$ with respect to the basis $\mathcal{B}$).
- $S_{\mathcal{B}\rightarrow\mathcal{C}}$ (a.k.a. the change-of-basis matrix from $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_d)$ to $\mathcal{C}=(\vec{w}_1,\dots,\vec{w}_d)$).
- $\left\lbrack T\right\rbrack_{\mathcal{B},\mathcal{C}}$ (a.k.a. the matrix of $T$ with respect to the bases $\mathcal{B}$ and $\mathcal{C}$).
Carefully state the following theorems (you do not need to prove them):[edit]
- Formula for $S_{\mathcal{B}\rightarrow\mathcal{C}}$ (in terms of the reduced row-echelon form of $\left\lbrack\vec{w}_1|\dots|\vec{w}_d|\vec{v}_1|\dots|\vec{v}_d\right\rbrack$).
- Transformation law for vectors (i.e. the formula for $\left\lbrack\vec{x}\right\rbrack_{\mathcal{C}}$ in terms of $S_{\mathcal{B}\rightarrow\mathcal{C}}$ and $\left\lbrack\vec{x}\right\rbrack_{\mathcal{B}}$).
- Transformation law for matrices (i.e. the formula for $\left\lbrack T\right\rbrack_{\mathcal{D},\mathcal{E}}$ in terms of $\left\lbrack T\right\rbrack_{\mathcal{B},\mathcal{C}}$ and certain change-of-basis matrices).
Solve the following problems:[edit]
- Section 3.4, problems 19, 21, 23, 25, 27, and 37 (in problems 19-27, ignore the instructions and simply compute $\left\lbrack T\right\rbrack_{\mathcal{B},\mathcal{B}}$).
