Linear Transformations

A linear/matrix transformation is just a function. However, in linear algebra we use the term transformation because a function is like moving a vector.

A linear transformation is a transformation that keeps grid of space straight and origin constant.

Matrix of a Transformation §

The constants for vector’s components in linear combination of basis vectors remain the same for linearly transformed basis vectors.

All of this means that all you need is 4 numbers to represent linear transformation: the terminal point of $\hat{i}$ and the terminal point of $\hat{j}$ after the transformation. Given that $\hat{i}$ lands at $(a,b)$ and $\hat{j}$ lands at $(c,d)$, you can then package the transformation into a 2x2 matrix:

$$\begin{array}{r}\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\end{array}$$

This matrix is called the standard matrix.

To transform any vector $(x,y)$, you can just compute as follows:

$$x\cdot \left[\begin{array}{c}a\\ b\end{array}\right]+y\cdot \left[\begin{array}{c}c\\ d\end{array}\right]$$

Or in matrix-vector multiplication form:

$$\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

A good example of a transformation matrix with its columns as the new basis vectors are rotation matrices.

You can use the Inverse of a Matrix to find the reversing transformation.

One-to-one Transformation §

$T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is one-to-one if each $\vec{b}$ in ${\mathbb{R}}^m$ is the image of at most one $\vec{x}$ in ${\mathbb{R}}^n$. In other words, every $\vec{b}$ in ${\mathbb{R}}^m$ has a preimage in ${\mathbb{R}}^n$ such that $\vec{b}=T(\vec{x})$.

Or equivalently,

$$\mathbf{A}\vec{x}=0$$

has only $\vec{0}$ as the solution.

Given its transformation matrix $\mathbf{A}$, it must satisfy the condition that there are at least as many rows as columns, the columns are linearly independent, and there must be a pivot position in each column.

Also called an [[Single Variable Calculus/Functions#Injection||injective function]].

Onto Transformations §

$T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is onto if each $\vec{b}$ in ${\mathbb{R}}^m$ is the image of at least one $\vec{x}$ in ${\mathbb{R}}^n$. In other words, every $\vec{b}$ in ${\mathbb{R}}^m$ has a preimage in ${\mathbb{R}}^n$ such that $\vec{b}=T(\vec{x})$.

Given its transformation matrix $\mathbf{A}$, we know that the reduced form of $\mathbf{A}$ must have a pivot position in each row so that if you augment matrix $\mathbf{A}$ with a vector $\vec{b}$, there will be no constraints on the possible values of b in the rightmost column.

Also called an [[Single Variable Calculus/Functions#Surjection||onto function]].

Example Problems for Linear Transformations §

Problem 1 §

$$\mathbf{A}=\left(\begin{array}{cc}1& -3\\ 3& 5\\ -1& 7\end{array}\right),\vec{u}=\left(\begin{array}{c}2\\ -1\end{array}\right),\vec{b}=\left(\begin{array}{c}3\\ 2\\ -5\end{array}\right),\left(\begin{array}{c}3\\ 2\\ 5\end{array}\right)$$

**Calculate $T(\vec{u})=\mathbf{A}\vec{u}$

$$\mathbf{A}\cdot \vec{u}=\left(\begin{array}{c}5\\ 1\\ -9\end{array}\right)$$

Is $\vec{b}$ in the range of $T$? If it is, find its preimage $\vec{x}$.

$$\begin{array}{rl}\mathbf{A}\cdot \vec{x}& =\vec{b}\\ \left(\begin{array}{cc}1& -3\\ 3& 5\\ -1& 7\end{array}\right)\cdot \left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)& =\left(\begin{array}{c}3\\ 2\\ -5\end{array}\right)\end{array}$$ $$\left(\begin{array}{ccc}1& -3& 3\\ 3& 5& 2\\ -1& 7& -5\end{array}\right)$$ $$\begin{array}{rl}{R}_2=R_2-3R_1,R_3\to R_1+R_3& ⟹\left(\begin{array}{ccc}1& -3& 3\\ 0& 14& -7\\ 0& 4& -2\end{array}\right)\\ R_3\to R_3-\frac{2}{7}{R}_2,R_1\to R_1+\frac{3}{4}{R}_3& ⟹\left(\begin{array}{ccc}1& 0& \frac{3}{2}\\ 0& 1& -\frac{1}{2}\\ 0& 0& 0\end{array}\right)\end{array}$$ $$\vec{x}=\left(\begin{array}{c}\frac{3}{2}\\ -\frac{1}{2}\end{array}\right)$$

Is $\vec{c}$ in the range of $T$, and find the preimage if so.

$$\begin{array}{rl}\mathbf{A}\cdot \vec{x}& =\vec{c}\\ \left(\begin{array}{cc}1& -3\\ 3& 5\\ -1& 7\end{array}\right)\cdot \left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)& =\left(\begin{array}{c}3\\ 2\\ 5\end{array}\right)\end{array}$$ $$\left(\begin{array}{ccc}1& -3& 3\\ 3& 5& 2\\ -1& 7& -5\end{array}\right)$$

These 3 equations contradict one another.

Example Problems for Matrices §

Problem 1 §

Find the standard matrix A for the dilation transformation $T(\vec{x})=3\vec{x},\vec{x}\in {\mathbb{R}}^2$

$$\begin{array}{rl}{\vec{e}}_1& =\left[\begin{array}{c}3\\ 0\end{array}\right]\\ {\vec{e}}_2& =\left[\begin{array}{c}0\\ 3\end{array}\right]\\ \mathbf{A}& =\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]\end{array}$$

Problem 2 §

$T(\vec{x})$ rotates points about the origin with an angle of $\theta =\frac{-3\pi }{2}$.

$$\begin{array}{rl}T({\vec{e}}_1)& =\left[\begin{array}{c}0\\ 1\end{array}\right]\\ T(\overrightarrow{e_2})& =\left[\begin{array}{c}-1\\ 0\end{array}\right]\\ \mathbf{A}& =\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\end{array}$$

Problem 3 §

Find the standard matrix for the reflection across the plane x + y + z = 1.

Solution §

The transformation can be described as subtracting the normal unit vector of the plane multiplied by 2 times the magnitude of the displacement between the the point (vector) and the plane. To find displacement, we modify the distance formula between a point and a plane to not include the absolute value sign - this gives us the position of the point relative to the plane (above or below).

$$\begin{array}{rl}D& =\frac{ax+by+cz+d}{||\vec{n}||}\\ & =\frac{x+y+z-1}{\sqrt{3}}\\ \hat{n}& =\frac{<1,1,1>}{\sqrt{3}}\\ \vec{v}& \to \vec{v}-2D\cdot \hat{n}\\ <x,y,z>& \to <x,y,z>-2<\frac{x+y+z-1}{3},\cdots >\\ & \to <\frac{1}{3}x-\frac{2}{3}y-\frac{2}{3}z+\frac{2}{3},\cdots >\\ \mathbf{A}& =\left[\begin{array}{ccc}\frac{1}{3}& -\frac{2}{3}& -\frac{2}{3}\\ -\frac{2}{3}& \frac{1}{3}& -\frac{2}{3}\\ -\frac{2}{3}& -\frac{2}{3}& \frac{1}{3}\end{array}\right]\\ \mathbf{b}& =\left[\begin{array}{c}\frac{2}{3}\\ \frac{2}{3}\\ \frac{2}{3}\end{array}\right]\end{array}$$