Linear Algebra: Invertibility

What is Invertibility? What does that have to do with a matrix’s columns and shape?

• For a matrix to be invertible, it must represent a bijective transformation. In other words, for every vector in its codomain, there must be exactly one input vector that maps to it.

• The concept of range is relevant here, since, as its name suggests, the range $R(A)$ of a matrix $A$ is the set of vectors in the codomain that the matrix maps some input vector $\vec x$ to.

• So, for a matrix to be invertible, we want:

  • its range to be exactly equal to its codomain
  • every vector in the codomain must correspond to only one input vector $\vec x$ .

• In more "practical" terms, we should think of a matrix's range as the span of its columns. With this lens, we can say that, for an $m \times n$ matrix $A$ to be invertible:

  1. its columns should span $\mathbb{R}^m$ .
  2. its columns should form a linearly independent set of vectors.

For the first condition to be true, we need the matrix to have at least $m$ columns.

For the second condition to be true, given that each column of A is a vector in $\mathbb{R}^m$ , for all columns to form a linearly independent set, there cannot be more than $m$ columns.

• Therefore, for conditions 1. and 2. to be true, the matrix must have exactly $m$ columns; equal to the number of rows.

• This means that for matrix $A$ to be invertible, it must be a square matrix.