M.6 Range, Nullspace and Projections

Range of a matrix

The range of m × n matrix A, is the span of the n columns of A. In other words, for

$A = [a_{1} a_{2} a_{3} \dots a_{n}]$

where $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ are m-dimensional vectors,

$r a n g e (A) = R (A) = s p a n ({a_{1}, a_{2}, \dots, a_{n}}) = {v | v = \sum_{i = 1}^{n} c_{i} a_{i}, c_{i} \in R}$

The dimension (number of linear independent columns) of the range of A is called the rank of A. So if 6 × 3 dimensional matrix B has a 2 dimensional range, then $r a n k (A) = 2$ .

For example

$C = (\begin{matrix} 1 & 4 & 1 \\ - 8 & - 2 & 3 \\ 8 & 2 & - 2 \end{matrix}) = (\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}) = (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix})$

C has a rank of 3, because $x_{1}$ , $x_{2}$ and $x_{3}$ are linearly independent.

Nullspace

p>The nullspace of a m

\times

n matrix is the set of all n-dimensional vectors that equal the n-dimensional zero vector (the vector where every entry is 0) when multiplied by A. This is often denoted as

$N (A) = {v | A v = 0}$

The dimension of the nullspace of A is called the nullity of A. So if 6 $\times$ 3 dimensional matrix B has a 1 dimensional range, then $n u l l i t y (A) = 1$ .

The range and nullspace of a matrix are closely related. In particular, for m $\times$ n matrix A,

${w | w = u + v, u \in R (A^{T}), v \in N (A)} = R^{n}$

$R (A^{T}) \cap N (A) = ϕ$

This leads to the rank--nullity theorem, which says that the rank and the nullity of a matrix sum together to the number of columns of the matrix. To put it into symbols:

Rank--nullity Theorem

$A \in R^{m \times n} \Rightarrow r a n k (A) + n u l l i t y (A) = n$

For example, if B is a 4 $\times$ 3 matrix and $r a n k (B) = 2$ , then from the rank--nullity theorem, on can deduce that

$r a n k (B) + n u l l i t y (B) = 2 + n u l l i t y (B) = 3 \Rightarrow n u l l i t y (B) = 1$

Projection

The projection of a vector x onto the vector space J, denoted by Proj(X, J), is the vector $v \in J$ that minimizes $| x - v |$ . Often, the vector space J one is interested in is the range of the matrix A, and norm used is the Euclidian norm. In that case

$P r o j (x, R (A)) = {v \in R (A) | | x - v |_{2} \leq | x - w |_{2} \forall w \in R (A)}$

In other words

$P r o j (x, R (A)) = a r g m i n_{v \in R (A)} | x - v |_{2}$