Proving that Columns are Linearly Dependent

[B18]

Homework Statement

Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent.

Homework 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.

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 R^m and must be linearly dependent.

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

 

[Ray Vickson]

Homework Statement

Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent.

Homework 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.

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 R^m and must be linearly dependent.

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

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

 

[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.

 

[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.

 

[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.

 

[Ray Vickson]

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.

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