Bases and Linear Maps

dr4g0n9 · #1 $Old$ November 5th, 2009, 12:04 PM

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

tonio · #2 $Old$ November 5th, 2009, 12:23 PM

Quote:

Originally Posted by dr4g0n9 $View Post$

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

dr4g0n9 · #3 $Old$ November 5th, 2009, 12:25 PM

edited