Matrices as Transformations

Matrices as Transformations of the Plane #

The columns say where i and j land.

The columns of a 2 × 2 matrix are the places the unit vectors i and j land

Start with i, one step right, and j, one step up.

Column 1 of the matrix says where i lands: at (0, 1).

Column 2 says where j lands: at (−1, 0).

Those two columns together turn the whole plane a quarter turn.

Reflecting in the x-axis leaves i alone and sends j to the opposite side.

An enlargement sends i and j three times as far, and a shear slides j sideways.

The Determinant as an Area Scale Factor #

What a determinant does to every area.

The determinant of a transformation matrix is the factor by which every area is multiplied

The unit square has area 1. Doubling both directions makes it 2 by 2.

Its matrix has determinant 2 × 2 − 0 × 0 = 4, which is the new area.

A quarter turn moves the square without resizing it, and its determinant is 1.

A reflection has determinant −1: the area is the same, but the plane is flipped.

Here ad − bc = 0, so the square is squashed onto a line with no area left.

Composing Transformations by Multiplying Matrices #

One after another, read right to left.

Doing one transformation after another multiplies their matrices, with the one applied first on the right

Reflect the point first, then turn the result. Each new matrix is written on the left.

One matrix does both transformations: R M, worked out as an ordinary product.

Swap the order and M R comes out different, so the picture is different too.

R M and M R are not the same matrix, so the order you work in matters.

The matrix next to the column acts first, so a product reads right to left.

Invariant Points and Lines #

What a transformation leaves exactly where it was.

An invariant point is one a matrix sends to itself, found by solving that the image equals the point

A point is invariant when the transformation sends it back to exactly where it was.

Take the reflection in the x-axis and ask which points it leaves where they are.

The equations give y = 0, so every point on the mirror line itself stays put.

Everything off that line is thrown across it, and only the line survives untouched.

A line can map onto itself while its points slide along it. That is an invariant line.

A rotation about the origin fixes just one point, and the origin is always that point.

Matrices as Transformations of Space #

Three columns, for where i, j and k land.

The columns of a 3 × 3 matrix are where the three unit arrows i, j and k land

Space has three unit arrows: i along x, j along y and k along z.

Column 1 sends i to (0, 1, 0), swung a quarter turn round, just as in the plane.

Column 3 leaves k alone, so nothing moves along z: this is a turn about the z-axis.

Send k to its opposite and leave i and j: that reflects space in the plane z = 0.

Twos down the diagonal send all three arrows twice as far: an enlargement of space.

Doing one transformation after another still multiplies, and still reads from the right.

The Determinant as a Volume Scale Factor #

What a 3 × 3 determinant does to every volume.

The determinant of a 3 × 3 transformation is the factor by which every volume is multiplied

Start from the unit cube, whose volume is 1, and let a matrix act on space.

This one stretches x by 2, y by 3 and z by 4, each direction on its own.

The cube becomes a 2 by 3 by 4 box, and 24 unit cubes fit inside it.

Its determinant is 24 as well, and every volume is multiplied by that factor.

A determinant of −1 keeps every volume and turns space inside out instead.

Row 2 is twice row 1, so det B is 0 and the cube is flattened onto a plane.