# Linear maps

## Homomorphisms of vector spaces

### Linear maps

#### Homomorphisms between vector spaces

Homomorphisms map between algebras, preserving the underlying structure.

A homomorphism vetween vector space $$V$$ and vector space $$W$$ can be described as:

$$\hom (V, W)$$

Homomorphism between vector spaces must preserve the group-like structure of the vector space.

$$f(u+v)=f(u)+f(v)$$

The homomorphism must also preserve scalar multiplication.

$$f(\alpha v)=\alpha f(v)$$

A linear map (or function) is a map from one input to an output which preserves addition and scalar multiplication.

That is if function $$f$$ is linear then:

$$f(aM+bN)=af(M)+bf(N)$$

#### Alternative names for homomorphisms

Vector spaces homomorphisms are also called linear maps or linear functions.

### Homomorphisms form a vector space

If we can can show that scalars can act on morphisms, then we can shwn that morphisms on a vector space are themselves a vector space.

Scalars can act on morphisms, and so morphisms of vector spaces are themselves vector spaces.

#### Dimensions of homomorphisms

We can identify the dimensionality of this new vector space from the dimensions of the original vector spaces.

$$\dim (\hom(V, W))=\dim V \dim W$$

### The pseudo-inverse

The definition of the inverse is that:

$$MM^{-1}=I$$

$$M^{-1}M=I$$

We also have:

$$MM^{-1}M=M$$

$$M^{-1}MM^{-1}=M^{-1}$$

#### The inverse of a homomorphism

Generally we don’t have inverses of homomorphisms as the number of dimensions are different.

We can, however, find a matrix $$M^+$$ which satisfies:

$$MM^+M=M$$

$$M^+MM^+=M^+$$

This is the pseudo-inverse.

### Linear and affine functions

#### Linear maps

Linear maps can be written as:

$$v=Mu$$

These go through the origin. That is, if $$u=0$$ then $$v=0$$.

#### Affine function

Affine functions are more general than linear maps. They can be written as:

$$v=Mu+c$$

Where $$c$$ is a vector in the same space as $$v$$.

Affine functions where $$c\ne 0$$ are not linear maps. They are not homomorphisms which preserve the structure of the vector space.

If we multiply $$u$$ by a scalar $$s$$, then $$v$$ will not increase by the same proportion.

### Singular value decomposition

The singular value decomposition of $$m\times n$$ matrix $$M$$ is:

$$M=U\Sigma V^*$$

Where:

• $$U$$ is a unitary matrix ($$m\times m$$)

• $$\Sigma$$ is a diagonal matrix with non-negative real numbers ($$m\times n$$)

• $$V$$ is a unitary matrix ($$n\times n$$)

$$\Sigma$$ is unique. $$U$$ and $$V$$ are not.

#### Properties

$$M^*M=U\Sigma^2 U^*$$

$$(M^*M)^{-1}=V\Sigma^{-2} V^*$$

#### Calculating the SVD

The SVD is generally calculated iteratively.

### Identity matrix and the Kronecker delta

#### The Kronecker delta

The Kronecker delta is defined as:

p$$\delta_{ij}=0$$ where $$i\ne j$$

$$\delta_{ij}=1$$ where $$i=j$$

We can use this to define matrices. For example for the identity matrix:

$$I_{ij}=\delta_{ij}$$

#### Identity matrix

A square matrix where every element is $$0$$ except where $$i=j$$. There is one for each square matrix.

$$I=\begin{bmatrix}1& 0&...&0\\0 & 1&...&0\\...&...&...&...\\0&0&...&1\end{bmatrix}$$