Math 260, Fall 2017, Assignment 9

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Read:

1. Section 3.4.
2. Section 5.1 (you may omit the material on correlation if you wish).

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

1. Linear transformation (from a subspace $S$ to a subspace $U$).
2. $[\vec{x}]_{\mathcal{B}}$ (i.e. the coordinate vector of $\vec{x}$ with respect to the basis $\mathcal{B}$).
3. $S_{\mathcal{B}\rightarrow\mathcal{C}}$ (i.e. the change-of-basis matrix from basis $\mathcal{B}$ to basis $\mathcal{C}$.
4. $[T]_{\mathcal{B},\mathcal{D}}$ (i.e. the matrix of a linear transofrmation $T:S\rightarrow U$ with respect to basis $\mathcal{B}$ for $S$ and basis $\mathcal{D}$ for $U$.
5. Orthonormal set.

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

1. Theorem concerning uniqueness of coordinates.
2. Linearity of coordinates.
3. Transformation law for vectors.
4. Transformation law for matrices.
5. Theorem characterizing orthonormality in terms of dot products.
6. Formula giving $[\vec{x}]_{\mathcal{B}}$ in terms of dot products, when $\mathcal{B}$ is orthonormal.
7. Formula for the components of $\vec{x}$ parallel and perpendicular to a subspace $S$, in terms of an orthonormal basis $\mathcal{B}=(\vec{u}_1,\dots,\vec{u}_k)$ for $S$.

Carefully describe the following algorithms:

1. Algorithm to compute $[\vec{x}]_{\mathcal{B}}$.
2. Algorithm to compute $S_{\mathcal{B}\rightarrow\mathcal{C}}$.

Solve the following problems:

1. Section 3.4, problems 1, 5, 7, 9, and 17.
2. Let $\mathcal{A}=(\vec{e}_1,\vec{e}_2)$ denote the "standard" basis for $\mathbb{R}^2$. For the data given in each of problems 19, 21, and 23 in section 3.4, let $\mathcal{B}=(\vec{v}_1,\vec{v}_2)$. Compute the change-of-basis matrices $S_{\mathcal{B}\rightarrow\mathcal{A}}$ and $S_{\mathcal{A}\rightarrow\mathcal{B}}$. How are these matrices related? Finally, for the transformation $T(\vec{x})=A\vec{x}$, compute the matrix $[T]_{\mathcal{B},\mathcal{B}}$.
3. Section 5.1, problem 27 (see example 4 on page 206 for one method).

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