Unit 4 Continued

Transformations

Review

T is a linear transformation from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ if

1. $T(\overrightarrow{v_1}+\overrightarrow{v_2})=T(\overrightarrow{v_1})+T(\overrightarrow{V_2})$
2. $T(a\vec{v})=aT(\vec{v})$

Example:

Let A be an $m\times n$ matrix.

Define T from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ by $T\vec{v}=A\vec{v}$

Then T is a linear transformation

Theorem

Let T be a linear transformation from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$. Define $\left[\begin{array}{c}T\end{array}\right]$ to be the matrix whose i-th column is $T(e_i)$, where ${\vec{e}}_i$ is the i-th column of $I_N$

Then $T(\vec{v})=\left[\begin{array}{c}T\end{array}\right]\vec{v}$ for all $\vec{v}$ in ${\mathbb{R}}^n$

Theorem

Let T be a linear transformation then Range(T) = COL($\left[\begin{array}{c}T\end{array}\right]$)

$\left[\begin{array}{c}T\end{array}\right]$ = $\left[\begin{array}{ccc}\overrightarrow{w_1}& \overrightarrow{v_2}& \overrightarrow{w_3}\end{array}\right]$

T from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ is onto if Rank($\left[\begin{array}{c}T\end{array}\right]$) = $m$

NULL($\left[\begin{array}{c}T\end{array}\right]$) is the set of vectors $\vec{v}$ for which $T(\vec{v})=0$ since $T(\vec{v})=\left[\begin{array}{c}T\end{array}\right]\vec{v}$ called the kernel of T.

Onto Definition

A transformation is onto if, for every vector b in ${\mathbb{R}}^m$, the equation $T(x)=b$ has at least one solution x in ${\mathbb{R}}^n$.

Theorem

T is one-to-one if and only if NULL($\left[\begin{array}{c}T\end{array}\right]$) = $\vec{0}$

Suppose $T(\vec{v})=\vec{0}$ only for $\vec{v}=0$, suppose

Hence:

$\left[\begin{array}{c}T\end{array}\right]\vec{v}=\vec{0}$ Only for $\vec{v}=\vec{0}$ if Rank($\left[\begin{array}{c}T\end{array}\right]$) is equal to # of columns.

T is a transformation from ${\mathbb{R}}^3$ to ${\mathbb{R}}^4$, it cannot be onto.

If T is a transform from ${\mathbb{R}}^5$ to ${\mathbb{R}}^3$ then it cannot be 1-to-1. $\left[\begin{array}{c}T\end{array}\right]$ is $3\times 5$, so rank is at most 3.

Markov Chains

Shows calculations or movement between states

Transition Matrix

• The ij entry is the percentage of population moving from state j to i.

Corresponds with:

stateDiagram
  s1 --> s2: .2
  s2 --> s1: .1
  s2 --> s2: .3
  s1 --> s1: .5
  s1 --> s3: .3
  s3 --> s1: .25
  s3 --> s3: .6
  s2 --> s3: .6
  s3 --> s2: .15

Initial state vector has the initial distribution of the pop. Into the different states.

$A\overrightarrow{v_0}=\overrightarrow{v_1}$

Where A is the transition matrix, $\overrightarrow{v_0}$ is the initial state. This tells you the states after a certain amount of time.