Math 260, Fall 2019, Assignment 7

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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. Image (of a linear transformation).
2. Kernel (of a linear transformation).
3. Linear combination (of vectors $\vec{v}_1,\dots,\vec{v}_k$).
4. Span (of $\vec{v}_1,\dots,\vec{v}_k$).

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

1. Theorem relating images to subspaces.
2. Theorem relating kernels to subspaces.
3. Theorem relating spans to subspaces.

Carefully describe the following algorithms:

1. Span membership algorithm.

Solve the following problems:

1. Section 3.1, problems 5, 9, 13, 14, and 15.
2. Is the set $W=\left\{\begin{bmatrix}x\\y\end{bmatrix}:x+y=1\right\}$ a linear subspace of $\mathbb{R}^2$?
3. Is the set $W=\left\{\begin{bmatrix}x\\y\end{bmatrix}:x+y=0\right\}$ a linear subspace of $\mathbb{R}^2$?
4. Is the set $W=\left\{\begin{bmatrix}x\\y\end{bmatrix}:x\leq y\right\}$ a linear subspace of $\mathbb{R}^2$?
5. Decide whether the vector $\begin{bmatrix}2\\1\\2\\1\end{bmatrix}$ lies in the plane spanned by $\begin{bmatrix}1\\0\\1\\0\end{bmatrix}$ and $\begin{bmatrix}1\\2\\1\\2\end{bmatrix}$.
6. Decide whether the vector $\begin{bmatrix}2\\1\\3\\2\end{bmatrix}$ lies in the plane spanned by $\begin{bmatrix}1\\0\\1\\0\end{bmatrix}$ and $\begin{bmatrix}1\\2\\1\\2\end{bmatrix}$.

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