# Bases and Linear Maps

• November 5th 2009, 10:04 AM
dr4g0n9
Bases and Linear Maps
Hi, Quick question:

If I know that T:V -> W is injective and B is a basis of V, how can i use that to show that T(B) is a basis of W?

Thanks
• November 5th 2009, 10:23 AM
tonio
Quote:

Originally Posted by dr4g0n9
Hi, Quick question:

If I know that T:V -> W is injective and B is a basis of V, how can i use that to show that T(B) is a basis of W?

Thanks

You can't because it isn't true: $T: \mathbb{R}\rightarrow \mathbb{R}^2\,,\,\,T(r)=(r,0)$ is injective but not onto and thus no basis of the definition set can be mapped to a basis of the image set.

What is true is that if T is injective then the image of a basis is a lin. indep. set.

Tonio
• November 5th 2009, 10:25 AM
dr4g0n9
edited