# A question on linear maps

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?

tiny-tim
Homework Helper
hi kostas230!
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?

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 ?)

i] that's a map from V to V, not to W

Silly me, I overlooked it xD

ii] what do you mean by V-U ? (eg what is R3 - R2 ?)

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

tiny-tim
Homework Helper
I mean the elements of V that do not belong in U.

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

tiny-tim
Homework Helper
Well, the correct answer would be: ℝ^3-ℝ^2={(0,0,z):z is real

that's a line

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

HallsofIvy
R3- R2 is all (x, y, z) such that $z\ne 0$ which is not a subspace.