Math 115AH Homework # 2

(additional problems)

Problem 1   Let $V$ be a vector space which is spanned by a finite set $S$ consisting of $n$ elements. Show that $V$ is finite-dimensional with dimension less then or equal to $n$. (You should not assume from the beginning that $V$ is finite-dimensional!)

Problem 2   a) Let $f_{1}(x)=1,\, f_{2}(x)=e^{ix},\, f_{3}(x)=e^{-ix}$. Prove that these functions are linearly independent in the space $\mathcal{F}(\mathbb{R},\mathbb{C})$ of all complex-valued functions on the real line.
b) Let $g_{1}(x)=1,$   $g_{2}(x)=\textrm{cos}(x),$   $g_{3}(x)=\textrm{sin}(x)$. Find an invertible $3\times 3$ matrix such that $g_{i}(x)=\sum _{j=1}^{3}P_{ij}(x)f_{j}(x)$ for all $1\leq i\leq 3$.

Problem 3   Let $V$ be a $4$-dimensional vector space, and $W$ be its $2$-dimensional subspace. Show that there is a $3$-dimensional subspace $U$ of $V$ such that $W\subset U\subset V$.

Problem 4   Let $V=P_{3}(\mathbb{R})$, $W=M_{2\times 2}(\mathbb{R})$ be two vector spaces, both of dimension $4$. Construct an explicit isomorphism from $V$ to $W$.

Problem 5   a) Prove that a linear map $T:V\to W$ is an isomorphism iff it sends a basis of $V$ to a basis of $W$.

b) Let $V$ and $W$ be vector spaces, and let $\textrm{dim}(V)=n$. Let $\{e_{1},\dots ,e_{n}\}$ be a basis in $V$ and $\{y_{1},\dots ,y_{n}\}$ be an arbitrary set of vectors in $W$. Prove (by giving an explicit construction) that there exists a linear transformation $T:V\to W$ such that $T(e_{i})=y_{i}$ for all $i=1,\dots ,n$. Is such a transformation unique?

Problem 6   (Annihilators) Let $S$ be a subset of a vector space $V$.

1. Prove that its annihilator $S^{0}=\{h\in V^{*}:\, \langle h,x\rangle =0\, \forall x\in S\}$ is a subspace of $V^{*}$.
2. Prove that if $S\subset V$ is a subspace and $v\notin S$, then there exists $h\in S^{0}$ such that $h(v)\neq 0$.
3. Let $V$ be finite-dimensional and $\psi :V\to V^{**}$ be the isomorphism defined by $x\mapsto \hat{x}$, where $\hat{x}(h)=h(x)$ for all $h\in V^{*}$. Prove that $(S^{0})^{0}=\textrm{span}(\psi (S))$. In particular, if $S$ is a subspace, conclude that $S^{00}=\psi (S)$ (i.e., $S^{00}$ and $S$ can be identified by the natural isomorphism between $V$ and $V^{**}$).
4. Let $W_{1}$ and $W_{2}$ be subspaces. Show that $(W_{1}+W_{2})^{0}=W_{1}^{0}\cap W_{2}^{0}$ and $(W_{1}\cap W_{2})^{0}=W_{1}^{0}+W_{2}^{0}$. (Hint: first prove that $U\subset W$ implies that $W^{0}\subset U^{0}$).

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Math 115AH Homework # 2

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