Vector Spaces

Rank from RREF

Find the rank of a matrix by row-reducing it to row echelon form (REF) and counting pivot columns. The number of free variables equals the number of columns minus the rank.

rank The rank of a matrix is the number of pivot columns in its REF. It equals the dimension of the column space. The nullity (number of free variables) satisfies rank + nullity = number of columns.

Example

Reduce and count pivot rows.

highlighted = computed this step

Step 1 — Set up reduction

Set up the augmented matrix for row reduction.

R=[121243364]R=\left[\begin{array}{cc|c}1&2&1\\2&4&3\\3&6&4\end{array}\right]

Step 2 — Update row 2

Row-reduce with R2 <- R2 - 2*R1.

R=[121001364]R=\left[\begin{array}{cc|c}1&2&1\\\hl{0}&\hl{0}&\hl{1}\\3&6&4\end{array}\right]

Step 3 — Update row 3

Row-reduce with R3 <- R3 - 3*R1.

R=[121001001]R=\left[\begin{array}{cc|c}1&2&1\\0&0&1\\\hl{0}&\hl{0}&\hl{1}\end{array}\right]

Step 4 — Update row 3

Row-reduce with R3 <- R3 - 1*R2.

R=[121001000]R=\left[\begin{array}{cc|c}1&2&1\\0&0&1\\\hl{0}&\hl{0}&\hl{0}\end{array}\right]

Step 5 — State verdict

rank = 2 (two pivot columns).

rank⁡=2\operatorname{rank}=\hl{2}