# Linear independence of columns of a matrix

## Homework Statement

Are the columns of this matrix linearly independent?
1........3.........-2
0.......-8.........11
0........0.........1
0....... 0......... 0
(periods are just to make spacing clear)

## The Attempt at a Solution

What is confusing me is the last row of zeros. If a set of vectors contains the zero vector, it is linearly dependent..but would this affect the linear independence of the columns of the matrix? If you augment the matrix with the zero vector, then the third row says that the only solution is the trivial one, which means that the columns of the matrix are linearly independent.

cristo
Staff Emeritus
If the columns are linearly dependent, then the third column can be made by adding multiples of the first and second columns together. Can this be done?

No it can't, so the fact that there is a row of zeros doesn't matter for the columns of the matrix..

cristo
Staff Emeritus