Proposition (Set of Linear maps is vector space)

The set of linear maps $$\mathscr{L}(V;W):=\{T:V\to W|\ T\text{ is linear}\}$$is a vector space.

Proposition (Operators have at most $dim(V)$ distinct eigenvalues)

Any operator $T\in\mathscr{L}(V)$ has at most $dim(V)$ distinct eigenvalues.

Proposition (Properties of linear maps)

1. Associative: Let $T_1\in\mathscr{L}(V_1,V_2),T_2\in\mathscr{L}(V_2,V_3)$ and $T_3\in\mathscr{L}(V_3,V_4)$. The composition $T_1\circ T_2\circ T_3$ is associative, meaning $$(T_1\circ T_2)\circ T_3 = T_1\circ (T_2\circ T_3)$$
2. Identity: If $T\in\mathscr{L}(V,W)$ then $$I_W\circ T = T = T\circ I_V$$
3. Distributivity: If $S_1\in\mathscr{L}(V_1,V_2),T_1,T_2\in\mathscr{L}(V_2,V_3)$, and $S_2\in \mathscr{L}(V_3,V_4)$ then $$(T_1+T_2)\circ S_1=T_1\circ S_1 + T_2\circ S_1\text{ and } S_2\circ (T_1 + T_2) = S_2\circ T_1 + S_2\circ T_2$$

Proposition (Linear maps are defined on a basis)

Suppose that ${v_1,...,v_n}$ is a basis for $V$ and $w_1,...,w_n\in W$. Then there is a unique linear map $T:V\to W$ such that $$T(v_j)=w_j$$ for $j=1,...,n$.

Proposition (Dimensionality & Linear Maps)

Let $V,W$ be Vector Spaces and let $T\in\mathscr{L}(V,W)$ be a Linear Map. Then

1. If $dim(W)<dim(V)$, then T is not Injective
2. If $dim(V)<dim(W)$, then T is not Surjective

Proposition (The inverse to a linear map is unique)

If a linear map $T\in\mathscr{L}(V,W)$ is invertible, the the inverse to $T$ is unique.

Proposition (3.14)

For $A\in \mathscr{L}(\mathbb{R}^{n};\mathbb{R}^{m})$, the following statements hold and are equivalent:

4. $A$ is continuous with respect to the standard topologies on $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$;
5. $A$ is continuous at $\mathbf{0}\in \mathbb{R}^{m}$;
6. there exists $M>0$ such that, for each $\boldsymbol{x}\in \mathbb{R}^{n}$, $$\lVert A(\boldsymbol x) \rVert_{\mathbb{R}^{m}}\le M\lVert \boldsymbol x \rVert _{\mathbb{R}^{n}} ;$$that is to say, $A$ is bounded.