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Up to now every matrix has been square: 2 by 2, 3 by 3. Those keep space the same size, 2D stays 2D. But a matrix does not have to be square. A nonsquare matrix takes vectors from one dimension and drops them into another. Once you read it by its columns, it is not scary at all.
Follow along with 3Blue1Brown: Nonsquare matrices as transformations between dimensions
Reading the shape of a matrix
A matrix has rows and columns. The shape is written as rows by columns.
- The number of columns tells you the input dimension (how many basis vectors you start with).
- The number of rows tells you the output dimension (how many numbers each landing vector has).
So a 3 by 2 matrix has 2 columns and 3 rows. Two columns means it eats 2D vectors. Three rows means each output is a 3D vector. It lifts 2D into 3D.
Columns are inputs, rows are outputs. A matrix that is "tall" (more rows than columns) lifts you into a bigger space. A matrix that is "wide" (more columns than rows) flattens you into a smaller one.
Lifting 2D into 3D
Take this 3 by 2 matrix:
| 2 0 |
| 1 1 |
| 1 -1 |
It has 2 columns, so it transforms 2D vectors. Each column is a 3D vector (3 numbers), so:
î = [1, 0] lands on [2, 1, 1] (the first column)
ĵ = [0, 1] lands on [0, 1, -1] (the second column)
You hand it a flat 2D vector, and it places it somewhere in 3D space. The result is that the whole 2D plane gets mapped onto a tilted, flat sheet (a plane) sitting inside 3D. The input was 2D and stays 2D in spirit; it just lives inside a bigger room now.
Applying it works the same as always, scale the columns and add:
| 2 0 | | 3 | | 2 | | 0 | | 6 |
| 1 1 | * | 1 | = 3 * | 1 | + 1 * | 1 | = | 4 |
| 1 -1 | | 1 | | -1 | | 2 |
A 2D input went in, a 3D output came out.
Flattening 3D into 2D
Now flip the shape. A 2 by 3 matrix has 3 columns and 2 rows:
| 3 1 4 |
| 1 5 9 |
Three columns means it eats 3D vectors. Two rows means each output is a 2D vector. So this one flattens 3D space onto the 2D plane.
î = [1,0,0] lands on [3, 1]
ĵ = [0,1,0] lands on [1, 5]
k̂ = [0,0,1] lands on [4, 9]
Three landing spots, each a flat 2D vector. This is exactly what happens when a 3D scene is projected onto your flat screen: a 3D-to-2D transformation.
What about a 1 by N matrix?
A matrix with a single row, like [2 1], has 2 columns and 1 row. It takes a 2D vector and outputs a single number (a 1D vector). It maps the plane onto the number line.
This little case looks small, but it is the secret hero of the next chapter. A row vector turning 2D into one number is exactly what a dot product does, and that surprising link is called duality.
Why "no determinant" for nonsquare matrices
The determinant measures how area or volume scales when you stay in the same dimension. But a nonsquare matrix changes the dimension, so "how much did area scale" is not even a sensible question. That is why determinants are defined only for square matrices. Same reason there is no plain inverse: you cannot perfectly rewind a step that threw away a dimension.
Why this matters later
- Computer graphics constantly flattens 3D worlds to 2D screens with wide matrices.
- Machine learning layers are usually nonsquare: a layer might take 784 input pixels and output 128 features. That is a 128 by 784 matrix moving you from one dimension to another.
- Embeddings in AI map huge sparse inputs down into a small dense space, a flattening transformation.
Columns are inputs, rows are outputs. Tall matrices lift into higher dimensions, wide matrices flatten into lower ones. The column rule never changes: each column is where one basis vector lands.
Quick gotchas
Shape is rows by columns, but meaning is the reverse. Columns set the input size, rows set the output size.
No determinant, no plain inverse for nonsquare matrices. Those need a square shape.
The columns still must have as many numbers as there are output dimensions (rows). If they do not line up, the matrix is written wrong.
What you walked away with
- A nonsquare matrix moves vectors between dimensions.
- Columns = input dimension, rows = output dimension.
- Tall matrices lift up, wide matrices flatten down, and a single row sends a vector to a single number.
- Determinants and plain inverses only exist for square matrices.
Next up, Chapter 9: the dot product, the simplest way to multiply two vectors into a number, plus a surprising connection (called duality) to those single-row matrices we just met.