# Homework Help: Proving that Columns are Linearly Dependent

1. Mar 16, 2015

### B18

1. The problem statement, all variables and given/known data
Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent.

2. Relevant equations
Its obvious that for the columns to be linearly dependent they must form a determinate that is equal to 0, or if one of the column vectors can be represented by a linear combination of the other vectors.

3. The attempt at a solution
It seems like there has to be more shown to prove this statement, however this is what I have right now:
Let A be an m x n matrix, and let m < n.
Then the set of n column vectors of A are in Rm and must be linearly dependent.

Is this it? or do I need to state a theorem in here somewhere?

2. Mar 16, 2015

### Ray Vickson

Hint: what is the maximum dimensionality of the space spanned by the columns (regarded as column vectors)?

3. Mar 16, 2015

### B18

The maximum dimension of the space spanned would have to be m+1, correct? For example if the vectors were from R3 we would need 4 column vectors so that they were linearly dependent.

Last edited: Mar 16, 2015
4. Mar 16, 2015

### Izzy

Show that the reduced row echelon form of the mxn matrix will have at most m pivots. Then there are n-m columns without pivots, which can all be expressed as linear combinations of the columns with pivots. Since they are linear combinations of other columns, they are linearly dependent.

5. Mar 17, 2015

### B18

It would have been helpful if our professor explained what pivots were. Thanks though Izzy I'll make sense of what you explained and go from there.

6. Mar 17, 2015

### Ray Vickson

Are you telling me that you think 4 or more 3-component vectors can possibly span a space of dimension 4?