Help with a linear algebra problem

[kant]

1)

Any fuction T: V->W . V and W are vector spaces over the field of rational numbers. The fuction T is called additive if T(x+y)=T(x)+ T(y).

Proof that any function T from v to w are additive, then it must be a linear transformation.



2) Let T:V->W be linear

prove that if T is 1 to 1 IFF T carries linear indep subset in V to Linear, indep subset in W.

 

[JasonRox]

1)

Any fuction T: V->W . V and W are vector spaces over the field of rational numbers. The fuction T is called additive if T(x+y)=T(x)+ T(y).

Proof that any function T from v to w are additive, then it must be a linear transformation.



2) Let T:V->W be linear

prove that if T is 1 to 1 IFF T carries linear indep subset in V to Linear, indep subset in W.

1. Hint - Write the definition of a linear transformation right next to the question.

2. Hint - Assume it maps a lin. ind. subset in V to a lin. dep. subset to W.

That's only one direction, but now you have to go the other way too. (if and only if)