Operations with Linear Transformations

[hbweb500]

Homework Statement

Let $$T:U \rightarrow V$$ be a linear transformation, and let U be finite-dimensional. Prove that if dim(U) > dim(V), then Range(T) = V is not possible.

Homework Equations

dim(U) = rank(T) + nullity(T)

The Attempt at a Solution

I almost think there must be a typo in the book. For instance, let U be P4 (the space of polynomials degree 4 and lower), and let V be P2. Let T be the second derivative operator. Then the Range of T is V. This example is even printed earlier in the same book that I got this question from.

Otherwise, I see no reason why the Range(T) couldn't be V. The rank(T) could at most be dim(V), but that is no problem, because the nullity(T) could be anywhere from dim(U) to dim(U)-dim(V).

So, is this a typo? Or (maybe more likely) am I missing something obvious?

 

[Billy Bob]

Let $$T:U \rightarrow V$$ be a linear transformation, and let U be finite-dimensional. Prove that if dim(U) > dim(V), then Range(T) = V is not possible.

Definitely a typo. It should be <.