Thank you. Give an example of a linear transformation T: V rightarrow W of finite dimensional vector spaces that is one-to-one but that is not onto. Solution Let, V=R2,W=R3 T(x,y)=T(x,y,0) Let, T(x,y)=T(p,q) (x,y,0)=(p,q,0) x=p y=q HEnce transformation is one to one But clearly not onto as it only maps to vectors of the form (x,y,0) in R3.

1. Thank you. Give an example of a linear transformation T: V rightarrow W of finite dimensional
vector spaces that is one-to-one but that is not onto.
Solution
Let, V=R2,W=R3
T(x,y)=T(x,y,0)
Let, T(x,y)=T(p,q)
(x,y,0)=(p,q,0)
x=p
y=q
HEnce transformation is one to one
But clearly not onto as it only maps to vectors of the form
(x,y,0) in R3