24 - Linear Map

Linear Map

Let $V,W$ be vector spaces over $\mathbf{F}$
Map $T:V\to W$ is linear if

1. Preserves additive structure

$$T(v_1+v_2)=T(v_1)+T(v_2)\text{ for all }{v}_1,v_2\in V$$

2. Preserves scalar multiplication

$$T(\lambda v)=\lambda T(v)\text{ for all }v\in V\text{ and }\lambda \in \mathbf{F}$$

How to show a map is linear $T$ needs to satisfy

Map

$$T({\lambda }_1{u}_1+{\lambda }_2{u}_2)={\lambda }_{1}T(u_1)+{\lambda }_{2}T(u_2)$$

Basically a combination of the additive structure and scalar multiplication at the same time!

Preservation of $0$ in Linear Maps

Let $V,W$ be vector spaces over $\mathbf{F}$
Let $T:V\to W$ be a linear map then

$$T(0_V)=0_W$$

Proof

$$T(0_V)+T(0_V)=T(0_V+0_V)=T(0_V)$$

Hence

$$T(0_V)=0_W$$

Equivalent Properties of Linear Maps

Let $V,W$ be vector spaces over $\mathbf{F}$
Let $T:V\to W$

1. $T$ is linear (aka linear map)
2. $T(\alpha v_1+\beta v_2=\alpha T(v_1)+\beta T(v_2)$ for all $v_1,v_2\in V$ and $\alpha ,\beta \in \mathbf{F}$
3. For any $n\ge 1$ for $v_1,\cdots ,v_n\in V$ and ${\alpha }_1,\cdots ,{\alpha }_n\in \mathbf{F}$ then

$$T({\alpha }_1{v}_1+\cdots +{\alpha }_n{v}_n)={\alpha }_{1}T(v_1)+{\alpha }_{n}T(v_n)$$

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Projection Map

Let $V$ be a vector spaces over $\mathbf{F}$ with subspaces $U,W$ such that $V=U\oplus W$
For $v\in V$ there are unique $u\in U,w\in W$ such that $v=u+w$
Then $P$ is defined as

$$P:V\to V\text{ by }P(v)=w$$

Where $P$ is the projection of $V$ onto $W$ along $U$

Proof that $P$ is a linear map

Take $v_1,v_2\in V$ and ${\alpha }_1,{\alpha }_2\in \mathbf{F}$
There are $u_1,u_2\in U,w_1,w_2\in W$ such that

$$v_1=u_1+w_1\text{ and }{v}_2=u_2+w_2$$

With

$$\begin{array}{rl}{\alpha }_1{v}_1+{\alpha }_2{v}_2& ={\alpha }_1(u_1+w_1)+{\alpha }_2(u_2+w_2)\\ & =({\alpha }_1{u}_1+{\alpha }_2{u}_2)+({\alpha }_1{w}_1+{\alpha }_2{w}_2)\end{array}$$

where ${\alpha }_1{u}_1+{\alpha }_2{u}_2\in U$ and ${\alpha }_1{w}_1+{\alpha }_2{w}_2\in W$
Hence by uniqueness

$$P({\alpha }_1{v}_1+{\alpha }_2{v}_2)={\alpha }_1{w}_1+{\alpha }_2{w}_2={\alpha }_{1}P(v_1)+{\alpha }_{2}P(v_2)$$

Trace $A=(a_{ij})\in {\mathcal{M}}_{n\times n}(\mathbb{R})$

For

$$\text{trace}(A):=a_{11}+a_{22}+\cdots +a_{nn}$$

The sum of the entries on the main diagonal of $A$

Trace Linear Map

$$\text{trace}:{\mathcal{M}}_{n\times n}(\mathbb{R})\to \mathbb{R}$$

is a linear map

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