Math 260, Fall 2018, Assignment 8
From cartan.math.umb.edu
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:[edit]
- Section 3.1.
- Section 3.2.
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Kernel (of a linear transformation).
- Redundant (vector, in an ordered list of vectors).
Carefully state the following theorems (you need not prove them):[edit]
- Theorem relating kernels to linear subspaces.
Carefully describe the following algorithms:[edit]
- Algorithm to find a spanning set for a kernel.
- Subspace membership algorithm.
- Redundancy detection algorithm.
- Subspace containment algorithm.
- Subspace equality algorithm.
Solve the following problems:[edit]
- Section 3.1, problems 1, 5, 10, 13, 14, 15, 17, 19, and 21.
- Section 3.2, problems 11, 13, 15, 17, and 19.
- 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 $\mathrm{span}\left(\begin{bmatrix}0\\-1\\1\\-1\end{bmatrix},\begin{bmatrix}2\\-1\\3\\1\end{bmatrix},\begin{bmatrix}1\\5\\0\\1\end{bmatrix}\right)$.
- 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 $\mathrm{span}\left(\begin{bmatrix}0\\-1\\1\\-1\end{bmatrix},\begin{bmatrix}2\\-1\\3\\1\end{bmatrix},\begin{bmatrix}1\\5\\0\\1\end{bmatrix}\right)$.