A Linear Map (or transformation) is a function that maps one vector space onto another one.

We use T(v) to depict v with the specific transformation applied to it.

## Terms

f: X → Y

X is referred to as the

- Domain
- Argument of the Function

Y is referred to as the

- Codomain
- Function Value of X
- Image of X under F

Image: Set of all function values

Rank: The dimension of the image.

Kernel\Null Space: The set of all zeroes of T, where a zero is all v's within the domain for T(v) = 0.

Nullity: The dimension of the kernel.

## Necessary Conditions (for it being **Linear**)

(i.e. how to test)

### Multiplication Condition

We can 'stretch' (multiply by a scalar) and then rotate something, and this is the same as rotating and then 'stretching' (i.e. the two operations are commutative).

(1)### Addition Condition

Similarly if we add two vectors together and then rotate, this is the same as rotating the two and then adding them.

(2)### T(0) = 0

This is because T(0) = T(0v) = 0T(v) = 0

## Function Values (i.e. The Results)

(5)Because T(u) (where u is a vector in your space) depends upon the knowledge of the basis, we cannot know the function value (or result) of T(u) without knowledge of the basis.

## Linear Maps with Matrices

If $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and x is a vector in $\mathbb{R}^n$ then we can write $T(x) as A(x)$, where a is an *m* x *n* matrix.

We define the linear map function $T_A : \mathbb{R}^n \rightarrow \mathbb{R}^m \text{ as } T_A(x) = Ax \text { for } x \in \mathbb{R}^n$

The easy way of working this out is to take what T(x) becomes and put that in the matrix.

E.g. for

(8)## Images

The image of a linear map is the set of all function values.

The image of an m*n matrix is the subset of $\mathbb{R}^m$ where the b = Ax (for some x in $\mathbb{R}^n$)

## Matrix Nullity

The nullity of a matrix is the dimension of the kernel.

The nullity of A (nullity(A)) is:

- The maximum number of independent vectors in the solution of Ax = 0
- The number of parameters in the solution of Ax = 0
- The number of non-leading columns in an equivalent row-echelon form

The columns of a matrix A are linearly independent iff nullity(A) = 0 (i.e. if T(v) is never 0, they're linearly independent).

## Rank of a Matrix

The rank of a matrix (rank(A)) is:

- The number of linearly independent columns of A
- The number of leading columns in an equivalent row-echelon form

## Rank & Nullity

Rank(A) + Nullity(A) = number of columns in A.

or in other words:

Rank(T) + Nullity(T) = dim(V)

### Solutions for Ax = b, determined by Rank and Nullity

- If rank(A) ≠ rank([A|b]) then there is no solution
- If rank(A) = rank([A|b]) then:
- if nullity(A) = 0 there is a unique solution
- if nullity(A) > 0 then the general solution is of the form

$x = x+p + \lambda_1k_1 + ... + \lambda_vk_v, \text{ for } \lambda_1,...,\lambda_v \in \mathbb{R}$

(where the x's are any solutions, and the k's form the basis for the kernel).

## Example

Rotation Matrix is for rotating an angle 90^{o} clockwise.