# M.6 Range, Nullspace and Projections

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

$A = [ a_1 a_2 a_3 \ldots a_n ]$

where $$a_1 , a_2 , a_3 , \ldots ,a_n$$ are m-dimensional vectors,

$range(A) = R(A) = span(\{a_1, a_2, \ldots , a_n \} ) = \{ v| v= \sum_{i = 1}^{n} c_i a_i , c_i \in \mathbb{R} \}$

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 $$rank(A) = 2$$.

For example

$C =\begin{pmatrix} 1 & 4 & 1\\ -8 & -2 & 3\\ 8 & 2 & -2 \end{pmatrix} = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix}= \begin{pmatrix} y_1 \\ y_2\\ y_3 \end{pmatrix}$

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

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

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 $$nullity(A) = 1$$.

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

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

$R(A^T) \cap N(A) = \phi$

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

$A \in \mathbb{R}^{m \times n} \Rightarrow rank(A) + nullity(A) = n$

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

$rank(B) + nullity(B) = 2 + nullity(B) = 3 \Rightarrow nullity(B) = 1$

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 $$\vert x - v \vert$$.  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

$Proj(x,R(A)) = \{ v \in R(A) | \vert x - v \vert_2 \leq \vert x - w \vert_2 \forall w \in R(A) \}$

In other words

$Proj(x,R(A)) = argmin_{v \in R(A)} \vert x - v \vert_2$

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