Required reading: Section 2.2-2.4

Read ahead in the textbook, since we won't cover the material in the exact same order. In particular, we will treat the notion of linear isomorphism, and compositions of maps, a bit sooner. For your convenience, I've posted my course notes on Blackboard under `course materials'.

Material covered this week includes (but is not limited to): The vector space $\mathcal{L}(V,W)$, composition of linear maps $$\mathcal{L}(V,W)\times \mathcal{L}(U,V)\to \mathcal{L}(U,W),\ \ (T,S)\mapsto T\circ S$$ The definition of dual space $V^*=\mathcal{L}(V,F)$. Linear isomorphisms. The Theorem that if $\dim(V)=\dim(W)<\infty$, then the conditions `onto', `one-to-one', `isomorphism' are all equivalent, but that this is false if the dimensions are different or infinite. The Theorem that $T\in\mathcal{L}(V,W)$ is an isomorphism if and only if there exists $S\in\mathcal{L}(W,V)$ such that $S\circ T=T_V$ and $T\circ S=I_W$, and that for $\dim(V)=\dim(W)<\infty$ these two conditions are equivalent. Coordinate vectors, matrix representations of linear maps.