# range

range (span) of A^k

### Syntax

[X,dim]=range(A,k)

### Arguments

- A
real square matrix

- k
integer

- X
orthonormal real matrix

- dim
integer (dimension of subspace)

### Description

Computation of Range `A^k`

; the first dim rows of `X`

span the
range of `A^k`

. The last rows of `X`

span the
orthogonal complement of the range. `X*X'`

is the Identity matrix

### Examples

A=rand(4,2)*rand(2,4); // 4 column vectors, 2 independent. [X,dim]=range(A,1);dim // compute the range y1=A*rand(4,1); //a vector which is in the range of A y2=rand(4,1); //a vector which is not in the range of A norm(X(dim+1:$,:)*y1) //the last entries are zeros, y1 is in the range of A norm(X(dim+1:$,:)*y2) //the last entries are not zeros I=X(1:dim,:)' //I is a basis of the range coeffs=X(1:dim,:)*y1 // components of y1 relative to the I basis norm(I*coeffs-y1) //check

### Used Functions

The `range`

function is based on the rowcomp function
which uses the svd decomposition.

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