Homework Help: Proof involving linear transformation of a set of vectors

1. Sep 29, 2010

Lanthanum

1. The problem statement, all variables and given/known data
Let T:$$\Re^{n}$$$$\rightarrow$$$$\Re^{m}$$ and let S={u,v,w}$$\in$$$$\Re^{n}$$.

If S is linearly dependent, show that {T(u), T(v), T(w)} is also linearly dependent.

2. Relevant equations
N/A

3. The attempt at a solution
Since S$$\in$$$$\Re^{n}$$ then S$$\in$$$$\Re^{m}$$.
Not sure where to go from here
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 29, 2010

ZioX

If the set of vectors are dependent you can find a linear relationship between them.

What happens to this linear relationship under the map T?

3. Sep 29, 2010

Lanthanum

If p(t) is the linear relationship between the vectors in S, does that mean that T[p(t)] is the linear relationship between the vectors in S?

If so I guess that proves that S is also linearly dependent since there exists the linear relationship T[p(t)] between its elements.

4. Sep 30, 2010

ZioX

Look up the definition of linear dependence.

5. Sep 30, 2010

Something is lacking in your statement of the problem. Is T supposed to be a linear map? You did not say so.

6. Sep 30, 2010

HallsofIvy

Don't ask, work it out. Your "p(t) is the linear relationship" is what Ziox suggested you look up- the definition of "linearly dependent". If vectors, u, v, and w are linearly dependent, then there exist numbers, a, b, c, not all 0, such that au+ bv+ cw= 0.

Now look at T(au+ bv+ cw)= T(0).

As Arkajad suggested, you will need to specify that T is a linear transformation (which you do in a later post).

7. Sep 30, 2010

Lanthanum

Aha! I think I have it.

Since T is a linear transformation, T(0)=0, and T(au+ bv+ cw)=aT(u)+bT(v)+cT(w)=0

Since there exists scalars, a, b, c, not all 0, such that the above equation is true, S is linearly dependent.

I cant believe I missed that even after ZioX's advice, guess I needed a study break. Thanks guys.