Definitions

The following are all subspaces (which you should verify) associated with an mxn matrix A.

Column Space of A, Range of A, or Range Space of A, denoted R(A), is the set of all vectors b in R^m such that Ax=b for some x in R^n.

Note that this is also the set of all linear combinations of the columns of A.

Row Space of A, denoted R(A^T) is the Column Space of Transpose[A], that is, it is the set of all vectors f in R^n such that Transpose[A]y=f for some y in R^m.

Note that this is the set of all linear combinations of the rows of A.

Nullspace of A or Kernel of A, denoted N(A) or ker(A) is the set of all x in R^n such that Ax=0.

Left Nullspace of A, denoted N(A^T) is the nullspace of Transpose[A], that is, the set of all y in R^m such that Transpose[A]y=0.

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