Math 260, Fall 2019, 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

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

1. Redundant vector (in a list $\vec{v}_1,\dots,\vec{v}_k$).
2. Linear relation.
3. Trivial linear relation.
4. Linearly independent.

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

1. Theorem relating kernels to linear relations.
2. Theorem giving three additional conditions equivalent to linear independence.

Carefully describe the following algorithms:

1. Procedure to give a spanning set for an image.
2. Procedure to give a spanning set for a kernel.
3. Subspace containment test.
4. Subspace equality test.
5. Redundancy detection algorithm.

Solve the following problems:

1. Section 3.2, problems 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 42, 45, and 47.
2. Consider the subspaces $V=\mathrm{span}\left(\begin{bmatrix}1\\0\\-2\\3\end{bmatrix},\begin{bmatrix}1\\1\\1\\1\end{bmatrix}\right)$ and $W=\mathrm{span}\left(\begin{bmatrix}2\\1\\-1\\4\end{bmatrix},\begin{bmatrix}0\\-1\\-3\\2\end{bmatrix}\right)$. Is either of these contained in the other? Are they equal?
3. Repeat the above problem, leaving $V$ the same but replacing $W$ by $W=\mathrm{span}\left(\begin{bmatrix}2\\1\\-1\\4\end{bmatrix},\begin{bmatrix}1\\-1\\-3\\2\end{bmatrix}\right)$.
4. Repeat the above problem, leaving $V$ the same but replacing $W$ by $W=\mathrm{span}\left(\begin{bmatrix}2\\1\\-1\\4\end{bmatrix},\begin{bmatrix}4\\2\\-2\\8\end{bmatrix}\right)$.
5. Working in $\mathbb{R}^2$, how large is the largest linearly independent set of vectors you can make? What about in $\mathbb{R}^3$?
6. Working in $\mathbb{R}^2$, how many vectors do you need to span the whole space? What about in $\mathbb{R}^3$?

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