9.3: Review Problems

1. (Subspace Theorem) Suppose that $V$ is a vector space and that $U \subset V$ is a subset of $V$. Check all the vector space requirements to show that

$$\mu u_{1} + \nu u_{2} \in U \textit{ for all } u_{1}, u_{2} \in U, \mu, \nu \in \Re$$
implies that $U$ is a subspace of $V$.

2. (Subspaces spanning sets polynolmial span) Determine if $P_{3}^{\mathbb{R}}$ be the vector space of polynomials of degree 3 or less in the variable $x$.

$$x-x^{3} \in span\{ x^{2}, 2x+x^{2}, x+x^{3} \}.$$

3. (UandV) Let $U$ and $W$ be subspaces of $V$. Are:
a) $U\cup W$
b) $U\cap W$
also subspaces? Explain why or why not. Draw examples in $\Re^{3}$.

4. Let $L:\mathbb{R}^{3}\to \mathbb{R}^3$ where

$$L(x,y,z)=(x+2y+z,2x+y+z,0)\, .$$
Find ${\rm ker} L$, ${\rm im} L$ and eigenspaces $\mathbb{R}_{-1}$, $\mathbb{R}_{3}$. Your answers should be subsets of $\mathbb {R}^{3}$. Express them using the ${\rm span}$ notation.

Contributor

• David Cherney, Tom Denton, and Andrew Waldron (UC Davis)