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.

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.

Assignment #5 is due on Friday, October 28.

Additional homework (do not hand in).

• Section 2.3, Problems 1,2,3,13,16
• Section 2.4, Problems 4, 5, 9
• Give an example of a linear map $T\colon V\to V$, with $V\not=\{0\}$, such that $N(T)=R(T)$. (This is only possible if $\dim(V)$ is even -- why?)
• Prove that the composition of injective (=one-to-one) maps is again injective.
• Prove that the composition of surjective (=onto) maps is again surjective.
• Let $T:V\to W$ be injective (resp., surjective). Prove that $\dim V\le \dim W$ (resp., $\dim V\ge \dim W$).
• Let $V,W$ be vector spaces over $F$, and $S,T\colon V\to W$ two linear maps. Suppose $\dim W=1$. Prove that $S=c T$ for some non-zero scalar $c$ if and only if $N(S)=N(T)$.
• Let $T_1,T_2\colon V\to W$ be two linear maps with $R(T_1)=R(T_2)$, with $\dim V<\infty$. Prove that there exists an isomorphism $S\colon V\to V$ such that $T_2=T_1\circ S$. Is this also true if $\dim V=\infty$?
• Let $T_1,T_2\colon V\to W$ be two linear maps with $N(T_1)=N(T_2)$, with $\dim V<\infty$. Prove that there exists an isomorphism $R\colon W\to W$ such that $T_2=R\circ T_1$. Is this also true if $\dim V=\infty$?