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Rank and Null Space

Some matrices squash 3D space flat — onto a plane, or even a line. Rank counts the dimensions that survive; the null space is everything crushed to zero.

Step 1 of 40

Speed

The identity leaves the unit cube alone: three independent columns, and every point of 3D space is still reachable. Most matrices bend the cube into a slanted box. Some flatten it.

A =

rank = 3

nullity = 0

\operatorname{rref}(A) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

\operatorname{rank} A = 3, \qquad \operatorname{null} A = \{\mathbf{0}\}, \qquad 3 + 0 = 3

Rank and Null Space

Some matrices squash 3D space flat — onto a plane, or even a line. Rank counts the dimensions that survive; the null space is everything crushed to zero.

Step 1 of 40

Speed

The identity leaves the unit cube alone: three independent columns, and every point of 3D space is still reachable. Most matrices bend the cube into a slanted box. Some flatten it.

A =

rank = 3

nullity = 0

\operatorname{rref}(A) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

\operatorname{rank} A = 3, \qquad \operatorname{null} A = \{\mathbf{0}\}, \qquad 3 + 0 = 3

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Rank and Null Space

See also

Rank and Null Space

See also