No One-to-One Linear Transformation: V to W

[hannahlu92]

Homework Statement

Prove: If V and W are finite-dimensional vector spaces such that dim(W)<dim(V), then there is no one-to-one linear transformation T:V-->W

The Attempt at a Solution

I don't know how to do a well thought out proof.

 

[micromass]

Hi hannahlu92!

The first thing you should do with such a statement is trying to find concrete examples. Can you find examples of V and W such that dim(V)<dim(W). Is it true that there doesn't exist such a one-to-one map for these examples? (I.e. is it inuitively true).

Then, to actually start proving it, you'll need to unwind the concept. What does dimension mean? What does one-to-one mean? Can we find some connection between the definition of dimension and the concept of one-to-one maps?

 

[hannahlu92]

thank you for taking the time to try and help me. My final is tomorrow and I still can't understand Linear Algebra