Subspaces and Linear Functions

The range of every linear function f : Rn → Rm is a subspace of Rm , and every subspace of R m is the range of some linear function.

For this reason, subspaces of Rm are sometimes called linear spaces.

R(f) = R(A)= Span (A)

Column Space

That’s why R (A) is often called the column space of A.

Row Space

R (A^T) 

Strictly speaking, the range of AT is a set of columns, while the row space of A is a set of rows. However, no logical difficulties are encountered by considering them to be the same.

Equal Ranges

Nullspace

. N (f) is called the nullspace of f(some texts call it the kernel of f ), and it’s easy to see that N (f) is a subspace of Rn because the closure properties (A1) and (M1) are satisfified. 

R(x) = N(A)

Also

Computing P

Equal Nullspaces

Z