- Section 2.1, Problems 1, 2, 3, 21, 24, 25, 26, 27
- Section 2.2, Problems 1, 4, 13, 16
- Let $V$ be a vector space, $W_1,W_2$ two subspaces. Prove $ \dim(W_1+W_2)+\dim(W_1\cap W_2)=\dim(W_1)+\dim(W_2).$
- Let $F$ be a field, and $X$ a finite set. Show that the set of functions $\mathcal{F}(X,F)$, with addition and scalar multiplication defined `pointwise', is a vector space, and describe a basis of this vector space. Hint: Write $X=\{c_1,\ldots,c_n\}$ and follow the strategy from the Lagrange Interpolation Theorem. More generally, if $V$ is a vector space over $F$, show that $\mathcal{F}(X,V)$ is naturally a vector space, and that a basis of $V$ determines a basis of this vector space.

Given distinct $c_0,\ldots,c_n\in F$, there are unique polynomials $p_0,\ldots,p_n\in Pol_n(F)$ such that $p_i(c_j)=1$ if $i=j$, $=0$ otherwise. (These * Lagrange interpolation polynomials * are given by explicit formulas, which you must know.)

** Theorem: ** The Lagrange interpolation polynomials $p_0,\ldots,p_n$ form a basis of the vector space $Pol_n(F)$.

(Idea of proof: they are linearly independent, and there are $n+1$ of them.) Given $a_0,\ldots,a_n\in F$, there is a unique polynomial $p\in Pol_n(F)$ such that $p(c_i)=a_i$, given by the linear combination $p=a_0p_0+\ldots+a_np_n$. Make sure to practise this in concrete examples! (Note also that for finite fields, we must have $n<\# F$ in the discussion, since the $c_0,\ldots,c_n$ must be distinct.)