Subsection CSNM Column Space of a Nonsingular Matrix

Let us specialize to square matrices and contrast the column spaces of the coefficient matrices in Archetype A and Archetype B.

Example CSAA Column space of Archetype A
Example CSAB Column space of Archetype B

Example CSAA and Example CSAB together motivate the following equivalence, which says that nonsingular matrices have column spaces that are as big as possible.

Theorem CSNM Column Space of a Nonsingular Matrix

Suppose $A$ is a square matrix of size $n$. Then $A$ is nonsingular if and only if $\csp{A}=\complex{n}$.

Proof

With this equivalence for nonsingular matrices we can update our list, Theorem NME3.

Theorem NME4 Nonsingular Matrix Equivalences, Round 4

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 $\complex{n}$, $\csp{A}=\complex{n}$.

Proof
Sage NME4 Nonsingular Matrix Equivalences, Round 4
(in context)