# A question on linear maps

1. Aug 25, 2013

### kostas230

Suppose that V and W are finite dimensional and that U is a subspace of V. If dimU≥dimV-dimW prove that there exists a linear map T from V to W such that kerT=U.

Consider the following linear map:
T|u>=|u> if |u> belongs to V-U and T|u>=|0> if |u> belongs to U​

Therefore kerT=U
Is this correct?

2. Aug 25, 2013

### tiny-tim

hi kostas230!
nooo …

i] that's a map from V to V, not to W
ii] what do you mean by V-U ? (eg what is R3 - R2 ?)

3. Aug 25, 2013

### kostas230

Silly me, I overlooked it xD

I mean the elements of V that do not belong in U.

4. Aug 25, 2013

### tiny-tim

then R3 - R2 would include all elements with z ≠ 0

5. Aug 25, 2013

### kostas230

6. Aug 25, 2013

### tiny-tim

that's a line

add a line to R2 and you don't get R3

7. Aug 25, 2013

### HallsofIvy

Staff Emeritus
R2 direct sum a line gives R3 but that is not at all what you said!

R3- R2 is all (x, y, z) such that $z\ne 0$ which is not a subspace.
I think you need to review your definitions!