Math 260, Fall 2018, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Read:

1. Section 3.1.
2. Section 3.2.

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

1. Kernel (of a linear transformation).
2. Redundant (vector, in an ordered list of vectors).

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

1. Theorem relating kernels to linear subspaces.

Carefully describe the following algorithms:

1. Algorithm to find a spanning set for a kernel.
2. Subspace membership algorithm.
3. Redundancy detection algorithm.
4. Subspace containment algorithm.
5. Subspace equality algorithm.

Solve the following problems:

1. Section 3.1, problems 1, 5, 10, 13, 14, 15, 17, 19, and 21.
2. Section 3.2, problems 11, 13, 15, 17, and 19.
3. Determine whether $\mathrm{span}\left(\begin{bmatrix}1\\0\\2\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\-1\\3\\-1\end{bmatrix}\right)$ is a subset of $\mathrm{span}\left(\begin{bmatrix}0\\-1\\1\\-1\end{bmatrix},\begin{bmatrix}2\\-1\\3\\1\end{bmatrix},\begin{bmatrix}1\\5\\0\\1\end{bmatrix}\right)$.
4. Determine whether $\mathrm{span}\left(\begin{bmatrix}1\\0\\2\\0\end{bmatrix},\begin{bmatrix}1\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\-1\\3\\-1\end{bmatrix}\right)$ is equal to $\mathrm{span}\left(\begin{bmatrix}0\\-1\\1\\-1\end{bmatrix},\begin{bmatrix}2\\-1\\3\\1\end{bmatrix},\begin{bmatrix}1\\5\\0\\1\end{bmatrix}\right)$.

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