Overview

• Matrix rank

• Matrix nullity

• Rank-nullity theorem

• Row-echelon operations to determine rank, nullity

• Consider vectors $\left(\begin{array}{llll}{𝒂}_1& {𝒂}_2& \dots & {𝒂}_n\end{array}\right)=𝑨$ within the Euclidean vector space $({ℝ}^m,+,ℝ,\cdot )$. Ask: how much of ${ℝ}^m$ is within the range of $𝑨$?

Definition. The (column) rank of a matrix $𝑨$ is the dimension of its range, $ℛ\left(𝑨\right)$ (intutively, the number of linearly independent columns), $r=\mathrm{rank}\left(𝑨\right)=\dim ℛ\left(𝑨\right)$

Notes:

• $r⩽min(m,n)$, since there are $n$ vectors, and there can't be more linearly independent vectors than $m$, the dimension of the vector space ${ℝ}^m$

• The column rank is the same as the row rank or dimension of $ℛ\left({𝑨}^T\right)$

  1. The simplest proof is to consider $r>0$ as the smallest integer for which there exist matrices $𝑩\in {ℝ}^{m\times r}$, $𝑪\in {ℝ}^{r\times n}$ such that $𝑨=𝑩𝑪\Rightarrow {𝑨}^T={𝑪}^T{𝑩}^T$, and interpret $𝑩$ as the minimal spanning set of $𝑨$, and ${𝑪}^T$ as the minimal spanning set of ${𝑨}^T$. Both have $r$ vectors. If no such positive integer exists then $𝑨=𝟎$ of rank 0.

  ¹ .

• Recall that the null space of matrix $𝑨$ is $𝒩\left(𝑨\right)=\{𝒙|𝑨𝒙=𝟎.\}$.

Definition. The nullity of a matrix $𝑨$ is the dimension of its null space, $𝒩\left(𝑨\right)$ (intutively, the dimension of the space that cannot be reached by linear combination), $z=\mathrm{null}\left(𝑨\right)=\dim 𝒩\left(𝑨\right)$

The rank-nullity theorem essentially states that a vector can either be reached or not reached by a linear combination, $r+z=n$, there are no other possibilities.

• Reduction to row-echelon form can be used to determine matrix rank and nullity. Allowed operations:

  • multiply a row by a scalar

  • interchange rows

  • add a row to another row

• The objective is to form ones on the diagonal. The number of ones is the rank of the matrix.