One-dimensional Linear Algebra

Friday July 24, 2026

Linear Algebra is the mathematics of linear transformations. It can handle lots of dimensions, but let's explore the one-dimensional case with real numbers.

A linear transformation is called linear because it behaves like a straight line that goes through the origin.

In one dimension, linear transforms are just multiplication. For example, let's consider a transform \( M \) that multiplies by 4. We can write the matrix representation \( M = \begin{bmatrix} 4 \end{bmatrix} \).

To undo \( M \), we divide by 4, or equivalently multiply by the multiplicative inverse of 4, which is \( 4^{-1} = 0.25 \). The inverse of \( M \) is \( M^{-1} = \begin{bmatrix} 0.25 \end{bmatrix} \). So we have:

\[ M^{-1}M = MM^{-1} = \begin{bmatrix} 1 \end{bmatrix} = I \]

The transform \( I \) doesn't change anything and is called the identity.

Just as you can't divide by zero, not every transform has an inverse. The problem is when a transform sends multiple inputs to a single output. Such a transform is called singular and knowing its output is not enough to know what its input was. The determinant of a singular matrix is zero. (The determinant of \( M \) is 4.)

In one dimension the only singular transform is \( \begin{bmatrix} 0 \end{bmatrix} \). It's a one-dimensional transform, but its output is really zero-dimensional, and we say it has rank 0. In contrast, \( M \) has rank 1, which is full rank for a one-dimensional transform. As long as \( M \) is full rank, it has an inverse and we can solve for \( X \):

\[ \begin{align} MX &= C \\ X &= M^{-1}C \end{align} \]

A linear transformation can reorient inputs in a rigid way, rotating and flipping them without changing their distance to the origin. Transforms that only do this are called orthogonal and in our case there are only two: \( \begin{bmatrix} 1 \end{bmatrix} \) and \( \begin{bmatrix} -1 \end{bmatrix} \).

A linear transformation can also scale inputs without rotating them. In one dimension we only have one direction, and any vector \( v \) along that direction is just multiplied: \( Mv = \lambda v \). So we have a single eigenvalue, \( \lambda = 4 \), and any non-zero vector is a corresponding eigenvector. Often we choose an eigenvector with length one, like \( v = \begin{bmatrix} 1 \end{bmatrix} \).

Any linear transform can be expressed as reorientation followed by non-negative scaling followed by another reorientation, which is called a Singular Value Decomposition:

\[ M = U \Sigma V^T = \begin{bmatrix} 1 \end{bmatrix} \begin{bmatrix} 4 \end{bmatrix} \begin{bmatrix} 1 \end{bmatrix} \]

Singular values (in \( \Sigma \)) are always non-negative, while eigenvalues can be negative. If either are zero, the transformation is singular: some whole dimension is getting projected to zero.

In one dimension here, composing tranformations is just multiplying real numbers, which is commutative. This is lost as soon as we involve more dimensions: multiplying on the left is often different from multiplying on the right. There are some other additional complexities as well, but hopefully this one-dimensional tour is a helpful way to quickly survey some core ideas in an accessible way.