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}

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)
\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}