matrix spaces

Summary

Column space

• output space of the matrix, set of all vectors with solutions

the space that any vector in gets mapped to by the matrix

Relation to matrix multiplication

Rank-nullity theorem

Equivalent statements of invertibility

for a square matrix

Concept

For any given matrix

Row space

Basis of row space

• nonzero rows in rref
• since, two row equivalent matrices have the same rowspace

Column space

Basis of column space

• original columns that correspond to pivot columns in rref
• since, row operations preserve linear relations between columns

Rank

• dimension of the codomain space, how much does the matrix squish its original space
• preserved with transpose

Rank of augmented matrix

Nullspace

• solution space to the homogeneous system
• set of vectors that become null after transformation
• intersection of the linear equations when they are 0

nontrivial solutions to homogeneous system

Dimension of nullspace

the number of non-pivot columns or parameters in the general solution

Application

Spaces of a given matrix

Rowspace is orthogonal to nullspace

Check if vector is in rowspace, using nullspace