Notebook 5 – Math 2121, Fall 2020

In today's notebook we'll explore the plots of some linear transformations.

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Linear transformations

A linear transformation ${\mathbb{R}}^n\to {\mathbb{R}}^m$ is a function $T$ that takes inputs $x\in {\mathbb{R}}^n$ and produces outputs $T(x)\in {\mathbb{R}}^m$ such that $T(x+y)=T(x)+T(y)$ and $T(cx)=cT(x)$ for all $x,y\in {\mathbb{R}}^n$ and $c\in \mathbb{R}$.

If you have an $m\times n$ matrix $A$, then the formula $T(x)=Ax$ for $x\in {\mathbb{R}}^n$ defines a linear transformation ${\mathbb{R}}^n\to {\mathbb{R}}^m$. In this case we say that $A$ is the standard matrix of $T$.

Does every linear transformation arise in this way? (Yes.)

Can a linear transformation have more than one standard matrix? (No.)

Given a linear transformation, how do you find its standard matrix?

Let's investigate these questions when $n$ and $m$ are small numbers.

Helper methods

Creating a linear transformation from ${\mathbb{R}}^1\to {\mathbb{R}}^2$

Input in ${\mathbb{R}}^1$

x =

1

The pictures above show a random linear transformation $T:{\mathbb{R}}^1\to {\mathbb{R}}^2$.

When we vary the input value $x$, the output is a vector that always points along the same line.

Changing $x$ just rescales this vector.

If $A=T([1])$ then $T([x])=A[x]=xA$ for all $x\in \mathbb{R}$.

Creating a linear transformation from ${\mathbb{R}}^2\to {\mathbb{R}}^2$

Display: sum mesh

Input vectors in ${\mathbb{R}}^2$

x1 = y1 =

x2 = y2 =

rotate =

The pictures above show a random linear transformation $U:{\mathbb{R}}^2\to {\mathbb{R}}^2$.

The key property of this transformation is that it preserves parallelograms: the output of all points in the parallelogram with sides $x$ and $y$ is just the parallelogram with sides $U(x)$ and $U(y)$.

Another crucial aspect of linear transformations is that if $U_1:{\mathbb{R}}^m\to {\mathbb{R}}^p$ and $U_2:{\mathbb{R}}^n\to {\mathbb{R}}^m$ are both linear, then the function $U(x)=U_1(U_2(x))$ is a linear transformation $U:{\mathbb{R}}^n\to {\mathbb{R}}^p$.

We can test this for $n=m=p=2$.

Creating a linear transformation from a matrix

A_11 = A_12 =

A_21 = A_22 =

Input vectors in ${\mathbb{R}}^2$

x1 = y1 =

x2 = y2 =

rotate =

Nonlinear transformations

Finally, let's compare with plots of functions ${\mathbb{R}}^2\to {\mathbb{R}}^2$ that are not linear.

Input vectors in ${\mathbb{R}}^2$

x1 = y1 =

x2 = y2 =

rotate =

Note that parallelograms are not preserved by this transformation.