Proving Linear Dependence for Sets of Vectors in Rn

[bcjochim07]

Homework Statement

I'm starting to feel really hopeless about knowing how to approach any of these linear algebra proofs. :(

Give a "row vector proof" of the theorem that says "any set of m vectors in Rn is linearly dependent if m > n.

Homework Equations

The Attempt at a Solution

Here's my best try:

Let set {a1, a2, ... am} be a set of row vectors in Rn

These vectors can be represented in a matrix as follows:

|a1|
---
|a2|
---
...
|am|

Assume that m > n

am = c1a1 + c2a2 + ... c(m-1)a(m-1) ( I don't think I can make this statement)

The following row operations can be performed to get a zero row at the bottom:
Rm - c1R1 - c2R2 - ... c(m-1)R(m-1)


Therefore the vectors are linearly independent.

Could somebody help me fix this proof?

Here is the main problem I am having with this. When we treat them as row vectors and say that the number of vectors is greater than n, that is analagous to saying that we have more equations than unknowns. Now I know that you can say that if you have less equations than unknowns, that is a linearly dependent system, but I don't see how you can prove the opposite.

 

[HallsofIvy]

Homework Statement

I'm starting to feel really hopeless about knowing how to approach any of these linear algebra proofs. :(

Give a "row vector proof" of the theorem that says "any set of m vectors in Rn is linearly dependent if m > n.

Homework Equations

The Attempt at a Solution

Here's my best try:

Let set {a1, a2, ... am} be a set of row vectors in Rn

These vectors can be represented in a matrix as follows:

|a1|
---
|a2|
---
...
|am|

Assume that m > n

am = c1a1 + c2a2 + ... c(m-1)a(m-1) ( I don't think I can make this statement)

No, you can't. That's assuming the vectors are dependent, which is what you want to prove.

The following row operations can be performed to get a zero row at the bottom:
Rm - c1R1 - c2R2 - ... c(m-1)R(m-1)


Therefore the vectors are linearly independent.

Could somebody help me fix this proof?

Here is the main problem I am having with this. When we treat them as row vectors and say that the number of vectors is greater than n, that is analagous to saying that we have more equations than unknowns. Now I know that you can say that if you have less equations than unknowns, that is a linearly dependent system, but I don't see how you can prove the opposite.

You said you were to use a "row vector proof". Write out the definition of "dependent" and "independent" and you should see that you have m equations for n variables.

 

[bcjochim07]

Right, I see that there are m equations and n variables. So

if the vectors are linearly dependent
c1a1 + c2a2 + ... cmam = 0 where c1, c2, ... cm are not all zero.

but I'm still not understanding... we have m eqns. and n variables.

if m > n, how can you say anything about the solution to that system? Sure, if the m were < n, I could say, assuming that the system is consistent, the system has infinitely many solutions and the equations are linearly dependent. I'm still confused.

 

[Klockan3]

You are mixing up m and n, its "the numbers of vectors are m, the numbers of dimensions are n" is what that means.

Those aren't "m-vectors", but there are m vectors.