Linear Map

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Definition

LinearAlgebra

Let $V$ and $W$ be $\mathbb{F}$ -vector spaces. We define a linear map from $V$ to $W$ to be a function $T:V\to W$ such that 1. If $v_1, v_2\in V$ then $T(v_1+v_2)=T(v_1)+T(v_2)$
2. If $\lambda\in\mathbb{F}$ and $v\in V$ then $T(\lambda v)=\lambda T(v)$ .

A linear map is a function that “plays well” with addition and scalar multiplication. Even outside of linear algebra you should assume this to be true for anything deemed “linear”.

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

When $W=V$ and $T:V\to W$ , we call $T$ a Linear Operator, and we denote the vector space of linear operators on $V$ by $\mathscr{L}(V)$ .

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

Let $S,T\in\mathscr{L}(V,W), v\in V$ and $\lambda\in\mathbb{F}$ . The sum of $S$ and $T$ is defined to be the linear map $(S+T)(v)=S(v)+T(v)$ The product of $\lambda$ and $T$ is the linear map defined by $(\lambda T)(v)=\lambda(T(v))$

For $A\in \mathscr{L}(\mathbb{R}^{n};\mathbb{R}^{m})$ , the following statements hold and are equivalent: 1. $A$ is continuous with respect to the standard topologies on $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ ; 2. $A$ is continuous at $\mathbf{0}\in \mathbb{R}^{m}$ ; 3. 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.

Linear Map

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