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1. Let $W_1$ denote the set of all polynomials $f(x)$ in $P(F)$ such that in the representation
$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$
we have $a_i=0$ wheneven $i$ is even. Like wise Let $W_2$ denote the set of all polynomials $g(x)$ in $P(F)$ such that in the representation
$g(x)=b_mx^m+b_{m-1}x^{m-1}+...+b_1x+b_0$ ,
we have $b_i=0$ whenever $i$ is odd. Prove that $P(F)=W_1\oplus W_2$
2. Let $W$ be a subspace of a vector space $V$ over a field $F$. For any $v\in V$ the set $\left\{\,v\right\}+W=\left\{\,v+w : w\in W\right\}$ is called the $coset$ of $W containing$ v. It is customary to denote the coset by $v+W$ rather than ${v}+W$
(a) Prove that $v+W$ is a subspace of $V$ if and only if $v\in W$
(b) Prove that $v_1+W=v_2+W$ if and only if $v_1 - v_2\in W$
3. Let $f(x)$ be a polynomial of degree $n$ in $P_n(\mathbb{R} )$ Prove that for any $g(x)\in P_n(\mathbb{R} )$ there exist scalars $c_0,c_1,...,c_n$ such that
$g(x)=c_0f(x)+c_1f'(x)+c_2f''(x)+...+c_nf^{(n)}(x)$, where $f^{(n)}(x)$ denotes the $n$th derivative of $f(x)$
4. Let $v_1,v_2,...,v_k,v$ be vectors in a vector space $V$, and defind $W_1=span({v_1,v_2,...,v_k})$ , and $W_2=span({v_1,v_2,...,v_k,v})$.
(a) Find necessary and sufficient conditions on $v$ such that $dim(W_1)=dim(W_2)$
(b) State and prove a relationship involving $dim(W_1)$ and $dim(W_2)$ in the case that $dim(W_1)\not= dim(W_2)$
5. (a) Let $W_1$ and $W_2$ be subspaces of a vector space $V$ such that $V=W_1\oplus W_2$. If $\beta _1 $ and $\beta _2$ are bases for $W_1$ and $W_2$, respectively, show that $\beta _1\cap \beta _2=\varnothing $ and $\beta _1\cup \beta _2$ is a basis of $V$
(b) Conversely, let $\beta _1$ and $\beta _2$ be disjoint bases for subspaces $W_1$ and $W_2$, respectively, of a vector space $V$ Prove that if $\beta _1\cup \beta _2$ is a basis for $V$, then $V=W_1\oplus W_2$