Linear Maps

A linear map from $V$ to $W$ is a function $T:V\to W$ with the properties of:

Given any vectors $u$ and $v$

A linear map is a function $T$ that satisfies the following properties for any vectors $u$ and $v$ in a vector space $\mathbb{V}$ and any scalar $\lambda$:

• Additivity: $T(u+v)=Tu+Tv$
• Homogeneity: $T(\lambda v)=\lambda (Tv)$

In other words, a linear map preserves vector addition and scalar multiplication. Additionally, the composition of two linear maps is also a linear map, and the identity function is a linear map.

A linear map can be represented by a matrix in a specific basis, and the matrix of a linear map can be used to transform vectors from one basis to another.

Examples of linear maps are:

• The identity function $I:V\to V$ defined as $Iv=v$
• The differentiation operation $D:F(\mathbb{R})\to F(\mathbb{R})$ defined as $Df=f^{\prime }$
• The integration operation $T:F(\mathbb{R})\to \mathbb{R}$ defined as $Tf={\int }_0^{1}f(x)dx$
• A map $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ defined as $T(x_1,\dots ,x_n)=(A_{1,1}{x}_1+\cdots +A_{1,n}{x}_n,\dots ,A_{m,1}{x}_1+\cdots +A_{m,n}{x}_n)$

Example - The equation of a line

The general equation of a line

$$y=ax+b$$

Is not a linear map since it doesn’t have the homogeneity property:

$$f(\lambda x)\ne \lambda (f(x))$$

Because of the $b$ term, that in machine learning lingo is called bias.

Another example

Other equations may not be a linear map with respect to certain variables, but they can with respect to others.

For example the equation $y=z\sin (x)+z^2\sin (x)$ is not a linear map w.r.t. $z$ or $x$, but it is w.r.t. $\sin (x)$.

---

Linear maps form a vector space with addition and multiplication defined trivially. We also have the definition of product between linear maps.

Let $T:U\to V$ and $s:V\to W$ be two linear maps, their product $ST:U\to W$ is defined as:

$$(ST)(u)=S(Tu)$$

Keep in mind that composition of linear maps is not commutative: $ST\ne TS$.

Since linear maps for a vector space, they have also associativity, identity and distributive properties.