Linear Transformation

A Linear transformation is a Transformation where $T:{\mathbb{R}}^n\to {\mathbb{R}}^m,T(\vec{v})=A\vec{v}$

• where $A\in R^{m\times n}$
• Domain of T is ${\mathbb{R}}^n$ (where we start)
• Codomain or target of T is ${\mathbb{R}}^m$
• The vector $T(\vec{x})$ is the image of $\vec{x}$ under T
• The set of all possible images is called the range
• image $\in$ range $\in$ codomain
• When the domain and codomain are both $\mathbb{R}$, you can represent them as a Cartesian Graph in ${\mathbb{R}}^2$, as in a mapping of $\mathbb{R}\to \mathbb{R}$
  • If y is the codomain and x is the domain, the range is the range, the domain is all the images of f(x)
• Addition Rule
  • T(u + v) = T(u) + T(v)
• Multiplication Rule
  • T(c$\vec{v}$) = cT(v)
• Prove a transformation is linear by proving the addition and multiplication rules.
• “Principle of Superposition”
  • If we know $T(e_1)$, …, $T(e_n)$ then we know every T(v)

1-1

Injective

If there is at most one location in the Codomain for every location in the Domain

Link to original

• 1-1 $⟺$ every column of T is a pivot column
• 1-1 iff standard matrix has pivot in every column

Onto

Surjective

If there is a location in the Codomain for every location in the Domain

Link to original

• TLDR: Onto $⟺$ every row of T is a pivot row
• onto iff the standard matrix has a pivot in every row
• The matrix A has columns which span ${\mathbb{R}}^m$.
• The matrix A has all pivotal rows.

Bijective

Bijective

If there is exactly one location in the Codomain for every location in the Domain for the Transformation

Link to original

• Needs square matrix

Matrix Multiplication is a Linear Transformation

$$A=\left[\begin{array}{cc}1& 1\\ 0& 1\\ 1& 1\end{array}\right],\vec{u}=\left[\begin{array}{c}3\\ 4\end{array}\right],\vec{b}=\left[\begin{array}{c}7\\ 5\\ 7\end{array}\right]$$ $$T:{\mathbb{R}}^2\to {\mathbb{R}}^3$$

Compute $T(\vec{u})$

$$A\vec{u}=\left[\begin{array}{cc}1& 1\\ 0& 1\\ 1& 1\end{array}\right]\left[\begin{array}{c}3\\ 4\end{array}\right]=\left[\begin{array}{c}7\\ 4\\ 7\end{array}\right]$$

Calculate $\vec{v}\in {\mathbb{R}}^2$ so that $T(\vec{v})=\vec{b}$

$$A\vec{v}=\vec{b}$$ $$\left[\begin{array}{ccc}1& 1& 7\\ 0& 1& 5\\ 1& 1& 7\end{array}\right]\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 5\\ 1& 1& 7\end{array}\right]\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 5\\ 0& 1& 5\end{array}\right]\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 5\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 5\end{array}\right]$$ $$\vec{v}=\left[\begin{array}{c}2\\ 5\end{array}\right]$$

Give a $\vec{c}\in {\mathbb{R}}^3|\neg \exists \vec{v}|T(\vec{v})=\vec{c}$ $\vee$ Give a $\vec{c}$ that is not in the range of $T$ $\vee$ Give a $\vec{c}$ that is not in the span of the columns of $A$ (Same question for all)

Range of $T$ is a bunch of images of the following form:

$$A\vec{v}=\left[\begin{array}{c}{v}_1+v_2\\ v_2\\ v_1+v_2\end{array}\right]$$

For $\vec{c}$ to not be in the range of $T$, it cannot be in the above form, e.g. it can be

$$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$$