# Null: Null Spaces of Matrices In MASS: Support Functions and Datasets for Venables and Ripley's MASS

## 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

 `1` ```Null(M) ```

## 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))`.

## 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.

## References

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

`qr`, `qr.Q`.

## Examples

 ```1 2 3 4 5 6 7``` ```# 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] } ```

### Example output

```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]
}
```

MASS documentation built on May 3, 2021, 5:08 p.m.