Math 115AH Homework # 1

(additional problems)

Problem 1   a) Give an example of a subset of $\mathbb{R}^{2}$ which is closed under scalar multiplication, but is not closed under vector addition;

b) Give an example of a subset of $\mathbb{R}^{2}$ which is closed under vector addition, but not scalar multiplication;

Problem 2   Which of the following are subspaces of $C([-1,1])$:

1. $W=\{f\in C([-1,1]):\, f(-1)=f(1)\}$.
2. $W=\{f\in C([-1,1]):\, f(-1)=-f(1)\}$.
3. $W=\{f\in C([-1,1]):\, f(-1)=0\, \textrm{and}\, \textrm{f}(1)=0\}$.
4. $W=\{f\in C([-1,1]):\, f(-1)=0$ or $f(1)=0\}$.

Problem 3   Show that if $S_{1}$ and $S_{2}$ are subsets of a vector space $V$ with $S_{1}\subset S_{2}$, then $\textrm{span}(S_{1})\subseteq \textrm{span}(S_{2})$. In particular, if $S_{1}\subseteq S_{2}$ and $\textrm{span}(S_{1})=V$, then $\textrm{span}(S_{2})=V$.

Problem 4   a) Give an example of a vector space $V$ and two subspaces $W_{1}$ and $W_{2}$ such that $W_{1}\cup W_{2}$ is not a subspace;

b) Let $W_{1}$ and $W_{2}$ be subspaces of a vector space $V$. Give (and prove!) the necessary and sufficient condition for $W_{1}\cup W_{2}$ to be a subspace of $V$.

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Math 115AH Homework # 1

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