MATH1231 - Linear Maps

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)
\begin{equation} T(λv) = λT(v) \end{equation}

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

(2)
\begin{equation} T(u+v) = T(u) + T(v) \end{equation}
(3)
\begin{align} \text{e.g.} \end{align}
(4)
\begin{matrix} T \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix} & = T \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} + T \begin{pmatrix} 0 \\ 1 \\ 0 \\ \end{pmatrix} + T \begin{pmatrix} 0 \\ 0 \\ 1 \\ \end{pmatrix} \end{matrix}

### T(0) = 0

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

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

(5)
\begin{align} basis = \begin{Bmatrix} v_1 & v_2 & ... & v_n \end{Bmatrix} \end{align}
(6)
\begin{align} u = \begin {pmatrix} x_1 \\ x_2 \\ x_3 \\ \end {pmatrix} \end{align}
(7)
\begin{align} T(u) = x_1 \cdot T(v_1) + ... + x_n \cdot T(v_n) \end{align}

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)
\begin{matrix} T \begin{pmatrix} x_1\\ x_2\\ x_3\\ \end{pmatrix} = T \begin{pmatrix} 3x_1 & - & 5x_2 & + & 6x_3 \\ 0x_1 & + & 5x_2 & + & 31x_3 \\ \end{pmatrix} \end{matrix}
(9)
\begin{align} A = \begin {pmatrix} 3 & -5 & 6 \\ 0 & 5 & 31 \\ \end {pmatrix} \end{align}

## 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

1. If rank(A) ≠ rank([A|b]) then there is no solution
2. If rank(A) = rank([A|b]) then:
1. if nullity(A) = 0 there is a unique solution
2. 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 90o clockwise.

(10)
\begin{align} A_\alpha = \begin{pmatrix} cos(\alpha) & -sin(\alpha) \\ sin(\alpha) & cos(\alpha) \\ \end{pmatrix} \end{align}
page revision: 19, last edited: 25 Aug 2011 03:58