Linear Algebra - proof of transformation

[NewtonianAlch]

Homework Statement

Suppose T: V -> W is linear. Prove that T(0) = 0

The Attempt at a Solution

T(v) = Av
T(0) = A(0) = 0

Is that right?

 

[micromass]

What is A? Why can you write T(v)=Av? What is your definition of linear?

 

[NewtonianAlch]

A is the transformation matrix. For a transformation T, we need some kind of "transformer" and A is the transformation matrix I used. v is a random vector that is being transformed, in this case it's just the zero vector.

 

[HallsofIvy]

Essentially you are using what you are asked to prove- you can write a linear transformation as a matrix because, among other things, T(0)= 0.

A linear transformation, T, from vector space U to vector space V, must satisfy
1) T(u+ v)= T(u)+ T(v) with u and v vectors in U.
2) T(au)= aT(u) with a a member of the underlying field and u a vector in U.

Use (1) with v= 0 or (2) with a= 0.