Non-Square Matrices as Transformations Between Dimensions #

Hook: What About Transformations That Change Dimension? #

So far every transformation has been square – 2D to 2D, 3D to 3D. But matrices can have different numbers of rows and columns, representing transformations between spaces of different dimensions. A 3×2 matrix maps 2D vectors to 3D vectors; a 2×3 matrix maps 3D vectors to 2D vectors.

2D to 3D: A 3×2 Matrix #

A 3×2 matrix has three rows and two columns. The two columns tell you where the two input basis vectors land, and each landing spot requires three coordinates (the three rows).

[ \begin{bmatrix} 2 & 0\ -1 & 1\ -2 & 1 \end{bmatrix} ]

Here î lands at ((2, -1, -2)) and ĵ lands at ((0, 1, 1)). The output lives in 3D space, but the column space is a 2D plane through the origin – the span of those two vectors. Since the column space has the same number of dimensions as the input space, this matrix is full rank.

Geometrically: a flat 2D grid maps onto a tilted plane slicing through 3D space.

To apply the transformation to a 2D vector ((x, y)), scale each column by the corresponding coordinate and add as usual:

[ x \begin{bmatrix}2\-1\-2\end{bmatrix}

y \begin{bmatrix}0\1\1\end{bmatrix} = \begin{bmatrix}2x + 0y\-1x + 1y\-2x + 1y\end{bmatrix} ]

3D to 2D: A 2×3 Matrix #

A 2×3 matrix has two rows and three columns. The three columns indicate a 3D input space; the two rows mean each landing spot is described with two coordinates – the output is 2D.

[ \begin{bmatrix} a & b & c\ d & e & f \end{bmatrix} ]

Three basis vectors (î, ĵ, k̂) each land somewhere in 2D space. The transformation squishes all of 3D space onto a plane. This is always a lossy compression – the determinant is not defined for non-square matrices, but the rank tells you how many output dimensions survive.

2D to 1D: A 1×2 Matrix #

A 1×2 matrix has one row and two columns. It maps 2D vectors onto the number line (1D space):

[ \begin{bmatrix} a & b \end{bmatrix} ]

Each column has just a single entry – the number that basis vector lands at. For a vector ((x, y)):

[ x \cdot a + y \cdot b ]

This is a linear functional: it takes a 2D vector and produces a single number. It has a close relationship with the dot product, which will be explored next.

Geometrically, evenly spaced dots on a line in the input remain evenly spaced when mapped onto the number line.

Key Intuition #

Columns = number of input dimensions (basis vectors).
Rows = number of output dimensions (coordinates per landing spot).
The column space lives in the output space; its dimension is the rank.
Full rank means the column space dimension equals the input dimension.