Range, Rank, and Nullspace

The fundamental subspaces of a matrix (, column partition , see Matrix Algebra).

Range and Rank

• Range of ():
  • i.e. every possible (a linear combination of the columns)
• Rank:
  • = # of linearly independent columns of

Nullspace and Nullity

• Nullspace:
• Nullity:
  • = max # of linearly independent vectors in the nullspace

Rank-Nullity Theorem

Singular Matrices

• Fact: for ,
  • DNE (singular)
  • cols of not linearly independent
  • nontrivial

Nonsingular Matrices as a Basis

• Square, nonsingular matrices: given , . Then
  • the columns of provide a basis for
• any admits the expansion
  • are the “coordinates” of in the basis
• But then
• Any nonsingular matrix defines a basis for
  • This idea returns as state coordinate transformations