# Transformation using matrices

A vector could be represented by an ordered pair (x,y) but it could also be represented by a column matrix:

Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. This is called a vertex matrix.

Example

A square has its vertexes in the following coordinates (1,1), (-1,1), (-1,-1) and (1,-1). If we want to create our vertex matrix we plug each ordered pair into each column of a 4 column matrix:

We can use matrices to translate our figure, if we want to translate the figure x+3 and y+2 we simply add 3 to each x-coordinate and 2 to each y-coordinate.

If we want to dilate a figure we simply multiply each x- and y-coordinate with the scale factor we want to dilate with.

When we want to create a reflection image we multiply the vertex matrix of our figure with what is called a reflection matrix. The most common reflection matrices are:

for a reflection in the x-axis

for a reflection in the y-axis

for a reflection in the origin

for a reflection in the line y=x

Example

We want to create a reflection of the vector in the x-axis.

In order to create our reflection we must multiply it with correct reflection matrix

Hence the vertex matrix of our reflection is

If we want to rotate a figure we operate similar to when we create a reflection. If we want to counterclockwise rotate a figure 90° we multiply the vertex matrix with

If we want to counterclockwise rotate a figure 180° we multiply the vertex matrix with

If we want to counterclockwise rotate a figure 270°, or clockwise rotate a figure 90°, we multiply the vertex matrix with

**Video lesson:** Rotate the vector A 90° counter
clockwise and draw both vectors in the coordinate plane

**Circles, Basic information about circles**