nullity A = n - rank A \geq n - m > 0

. Thus a first dependent column prefix exists. Minimality of

k

makes the first

k - 1

columns independent, so

rank A_{k - 1} = k - 1

. Adding one column cannot reduce this rank, while nonzero nullity of

A_{k}

implies rank at most

k - 1

. Hence

rank A_{k} = k - 1, nullity A_{k} = 1.

For

k = 1

the earlier prefix is the zero-dimensional empty matrix, and the same argument applies.

 Ancestors (10)

 Discussion (0) New discussion

There are no discussions about this article yet.

There are currently no matching articles.