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

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Start with the unit vectors i, one unit right, and j, one unit up.

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

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

Together the two columns rotate the whole plane 90° counterclockwise about the origin.

Reflection in the x-axis leaves i where it is and sends j to (0, −1).

An enlargement by 3 sends i and j three times as far, and this shear moves j sideways to (2, 1).

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

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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 90° rotation moves the square without resizing it, and (0 −1; 1 0) has determinant 1.

A reflection has determinant −1: the area is unchanged, but the orientation is reversed.

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

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Reflect the point first, then rotate 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 combined transformation 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

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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 is invariant.

Every point off that line moves to the other side of it, so only the line is fixed.

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 vectors i, j and k land

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

Column 1 sends i to (0, 1, 0), a rotation of 90°, just as in the plane.

Column 3 leaves k where it is, so this is a rotation about the z-axis.

Send k to −k and leave i and j where they are: that is a reflection in the plane z = 0.

Twos down the main diagonal send all three unit vectors twice as far: an enlargement by 2.

Doing one transformation after another still multiplies, and the matrix on the right still acts first.

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

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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 but reverses orientation, as a reflection does.

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