MASS (version 7.3-58.2)

# Null: Null Spaces of Matrices

## Description

Given a matrix, `M`, find a matrix `N` giving a basis for the (left) null space. That is `crossprod(N, M) = t(N) %*% M` is an all-zero matrix and `N` has the maximum number of linearly independent columns.

## Usage

`Null(M)`

## Value

The matrix `N` with the basis for the (left) null space, or a matrix with zero columns if the matrix `M` is square and of maximal rank.

## Arguments

M

Input matrix. A vector is coerced to a 1-column matrix.

## Details

For a basis for the (right) null space \(\{x : Mx = 0\}\), use `Null(t(M))`.

## References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

`qr`, `qr.Q`.

## Examples

Run this code
``````# The function is currently defined as
function(M)
{
tmp <- qr(M)
set <- if(tmp\$rank == 0L) seq_len(ncol(M)) else  -seq_len(tmp\$rank)
qr.Q(tmp, complete = TRUE)[, set, drop = FALSE]
}
``````

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