Subsection BNM Bases and Nonsingular Matrices

A quick source of diverse bases for $\complex{m}$ is the set of columns of a nonsingular matrix.

Theorem CNMB Columns of Nonsingular Matrix are a Basis

Suppose that $A$ is a square matrix of size $m$. Then the columns of $A$ are a basis of $\complex{m}$ if and only if $A$ is nonsingular.

Proof
Example CABAK Columns as Basis, Archetype K

Perhaps we should view the fact that the standard unit vectors are a basis (Theorem SUVB) as just a simple corollary of Theorem CNMB? (See Proof Technique LC.)

With a new equivalence for a nonsingular matrix, we can update our list of equivalences.

Theorem NME5 Nonsingular Matrix Equivalences, Round 5

Suppose that $A$ is a square matrix of size $n$. The following are equivalent.

  1. $A$ is nonsingular.
  2. $A$ row-reduces to the identity matrix.
  3. The null space of $A$ contains only the zero vector, $\nsp{A}=\set{\zerovector}$.
  4. The linear system $\linearsystem{A}{\vect{b}}$ has a unique solution for every possible choice of $\vect{b}$.
  5. The columns of $A$ are a linearly independent set.
  6. $A$ is invertible.
  7. The column space of $A$ is $\complex{n}$, $\csp{A}=\complex{n}$.
  8. The columns of $A$ are a basis for $\complex{n}$.

Proof
Sage NME5 Nonsingular Matrix Equivalences, Round 5
(in context)