Linear Transformations

Imagine you have a set of points on a grid. Then a "transformation" maps (or moves) those points onto a new grid. Each point will transform to a new point. A Linear Transformation is a type of transformation that has certain properties, that will allow us to represent it with a matrix.

A diagram representing a transformation, T, from an xy-plane to another xy-plane. For example, the point (2,4) in the centre of the red circle is transformed to the point (2,3).

 

Prerequisites

We'll assume you already know about:

• Matrix Multiplication

Introductory Example

Figure 1. A square drawn onto an xy-coordinate grid, with vertices at O(0,0), A(4,-2), B(6,2) and C(2,4).

We will transform the square in Figure 1 with the matrix $T={\begin{pmatrix}1.5&-1\\0.5&1\end{pmatrix}}$.

In this topic, a linear transformation and a matrix are the same thing. The matrix T is the linear transformation T and the linear transformation T is the matrix T.

Transforming one Point

In order to transform point A:

• we'll first write point A as a position vector: ${\underline {a}}={\begin{pmatrix}4\\-2\end{pmatrix}}$
• we pre-multiply by the transformation matrix T: $T{\underline {a}}={\begin{pmatrix}1.5&-1\\0.5&1\end{pmatrix}}{\begin{pmatrix}4\\-2\end{pmatrix}}={\begin{pmatrix}8\\0\end{pmatrix}}$

Figure 2. The point A(4,-2) on an xy-coordinate grid is transformed, by the matrix T, to the point A*(8,0) on another xy-coordinate grid.

That's it! $T{\underline {a}}$ is the "transformed" ${\underline {a}}$. The point (4,-2) has been transformed to (8,0).

Object and Image

In order to avoid confusion, it's common to name the start point "the object A" and name the end point "the image of A". If we start with the object $A$, we may use $A^{*}$ to represent its image.

Transforming the Origin

One of the rules of Linear Transformations is that the origin won't move.

Notice: ${\begin{pmatrix}1.5&-1\\0.5&1\end{pmatrix}}{\begin{pmatrix}0\\0\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}$

The image of the origin is still the origin. In other words, in a linear transformation, the origin is always an invariant point.

Transforming Multiple Points at once

We could transform points A(4,-2), B(6,2) and C(2,4) one-by-one, by writing them each as position vectors ${\begin{pmatrix}4\\-2\end{pmatrix}},{\begin{pmatrix}6\\2\end{pmatrix}},{\begin{pmatrix}2\\4\end{pmatrix}}.$

Or, we could merge them into a single matrix: ${\begin{pmatrix}4&6&2\\-2&2&4\end{pmatrix}}.$

Then, pre-multiplying by the matrix T: ${\begin{pmatrix}1.5&-1\\0.5&1\end{pmatrix}}{\begin{pmatrix}4&6&2\\-2&2&4\end{pmatrix}}={\begin{pmatrix}8&7&-1\\0&5&5\end{pmatrix}}$

Working column-by-column, we see that:

• $A(4,-2)$ has transformed to $A^{*}(8,0)$, which we already found earlier,
• $B(6,2)$ has transformed to $B^{*}(7,5)$,
• $C(2,4)$ has transformed to $C^{*}(-1,5)$.

Outcome

Figure 3. The square OABC on an xy-coordinate grid is transformed, by the matrix T, to the parallelogram OA*B*C* on another xy-coordinate grid.

By drawing the new points, we see that the square OABC has been transformed onto the parallelogram OA*B*C*. It's sort of been both stretched and rotated.

"Linear" transformations generally transform straight lines onto new straight lines, so it's safe to join the new points with straight lines - in fact, the midpoint of AB will be transformed to the midpoint of A*B*.

Technical Detail

Technically, a linear transformation goes from a vector space to another vector space. At A-Level, the only vector spaces we know are coordinate grids, so we're considering only 2D and 3D coordinate grids.

Linear transformation have the properties that they preserve addition of vectors and preserve scalar multiplication. In other words:

• If ${\underline {a}}+{\underline {b}}={\underline {c}}$, then $T{\underline {a}}+T{\underline {b}}=T{\underline {c}}$
• $T(k{\underline {a}})=k(T{\underline {a}})$

This should match our understanding of the algebra of matrices and vectors, making matrices perfect for representing linear transformations.

Next Steps

• Lines of Invariant Points, Invariant Lines
• Types of Linear Transformation
• Combinations of Linear Transformations
• Area and Volume in Linear Transformations