8. Linear Transformations

Question 1

Linear Transformation

Answer 1

A linear transformation between two vector spaces V and W is a map T:V->W such that the following hold

1. T(v1 + v2) = T(v1) + T(v2)
2. T(cv1) = cT(v1)

Question 2

F-Linear

Answer 2

Let U,V,W be F-Vector spaces and T:U→V and S:V→W be F-Linear transformations. Then SoT:U→W is F-Linear.

Question 3

Isomorphism of F-Vector Space

Answer 3

Let V and W be F-Vector Spaces. A F-Linear
transformation T: U→W is an isomorphism of F-Vector spaces if there exists a linear transformation T’:W→V with T’oT=I_v and ToT’:I_w. This T is called an inverse of T

Question 4

Uniqueness of Inverse

Answer 4

If T is an invertible linear transformation between vector spaces V and W then its inverse is unique