Vector Spaces of A

Null space, column space, row space — the four fundamental subspaces visible from a single matrix.

A
×
rank(A)
2
nullity
1
dim col(A)
2
Column space basis
Span of the pivot columns. Dimension = 2.
[121]\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}
[241]\begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix}
Row space basis
Nonzero rows of RREF(A). Dimension = 2.
[10−3]\begin{bmatrix} 1 & 0 & -3 \end{bmatrix}
[013]\begin{bmatrix} 0 & 1 & 3 \end{bmatrix}
Null space basis
Solutions to A·x = 0. Dimension = 1.
[3−31]\begin{bmatrix} 3 \\ -3 \\ 1 \end{bmatrix}