Proof involving linear transformation of a set of vectors

[Lanthanum]

Homework Statement

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.

Homework Equations

N/A

The Attempt at a Solution

Since S$$\in$$$$\Re^{n}$$ then S`$$\in$$$$\Re^{m}$$.
Not sure where to go from here

 

[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?

 

[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.

 

[ZioX]

Look up the definition of linear dependence.

 

[arkajad]

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

 

[HallsofIvy]

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.

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).

 

[Lanthanum]

Aha! I think I have it.

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)

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 can't believe I missed that even after ZioX's advice, guess I needed a study break. Thanks guys.