## SVD constructs orthonormal bases for the nullspace and range of a matrix.

## Columns of

*U*with corresponding non-zero*Wi*are an orthonormal basis for the range.## Columns of

*V*with corresponding zero*Wi*are an orthonormal basis for the nullspace.## Hence immediately have solution for

*A***x**= 0, i.e. the columns of*V*with corresponding zero*Wi*.