Basis for the image of a surjective linear map.

[snipez90]

Homework Statement

Let V and W be vector spaces over F, and let T: V -> W be a surjective (onto) linear map. Suppose that {v1, ..., v_m, u1, ... , u_n} is a basis for V such that ker(T) = span({u1, ... , u_n}). Show that {T(v1), ... , T(v_m)} is a basis for W.

Homework Equations

Basic properties of linear maps. Linear independence.

The Attempt at a Solution

I have already proven that {T(v1), ... , T(v_m)} spans W, which I thought would be harder than showing linear independence. But here is where I am confused. We have to show that {T(v1), ... , T(v_m)} is linearly independent.

Suppose (a_1)T(v1) + ... + (a_m)T(v_m) = 0 for (a_1), ... (a_m) in F. Then
T( (a_1)(v1) + ... + (a_m)(v_m) ) = 0 since T is linear. Now if we also suppose that (a_1)(v1) + ... + (a_m)(v_m) = 0, then clearly (a_1) = (a_2) = ... = (a_m) = 0 since the set {v_1, ... , v_m} is linearly independent.

But I think I'm confused when (a_1)(v1) + ... + (a_m)(v_m) =/= 0 (which is certainly possible right?). However, I have an idea and I think that in this case, we still get (a_1) = ... = (a_m) = 0?

 

[xaos]

well, if T(w)=0 then w is in the KerT, so we can write this as a linear combination of the u's.

you shouldn't need any more of a hint than this.

 

[snipez90]

Thanks xao, I think that was exactly what I had in mind (and I should have sorted out this case before I made the post). Basically then, setting a linear combination of the v's equal to a linear combination of the u's implies (by isolating 0 on one side) that all coefficients of vectors are 0.

 

[HallsofIvy]

Sorry, I misread the problem.

 

[snipez90]

But ker(T) = {0} if and only if T is injective, and surjectivity does not rule out the possibility that a nonzero vector is mapped to the 0 vector.

And just to clarify, the u's xao is talking about are also in the linearly independent set (consisting of v1 to v_m and u1 to u_n, sorry for not using latex). So currently my proof of linear independence looks like this:

Suppose

<br /> a_1T(v_1)+ a_2T(v_2)+ \cdot\cdot\cdot+ a_mT(v_m)= T(a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m)= 0

But kert(T) = span({u1, ... , u_n}), so we can write
a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m = b_1u_1+ b_2v_2+ \cdot\cdot\cdot+ b_nu_n.

Then
a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m = b_1u_1 - b_2v_2 - \cdot\cdot\cdot - b_nu_n = 0

and it follows that a_1 = a_2 = \cdot\cdot\cdot= a_m = 0.

 

[xaos]

i'm hoping that '=' was a typo in the last step.

it looks like the dimension of V is m+n (the v's plus u's) while the dimension of W is m, so a surjective map only needs m (the v's) of the m+n basis vectors. that leaves n basis vectors (the u's) spanning KerT to map onto the zero of W.

 

[snipez90]

Yeah sorry I really screwed up the latex on that last line. It should read

Then

<br /> a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m - b_1u_1 - b_2u_2 - \cdot\cdot\cdot - b_nu_n = 0<br />

so the conclusion follows.