Let V and W be vector spaces and T:V----->W a linear transformation. Then T is a

Question

Let V and W be vector spaces and T:V----->W a linear transformation. Then T is a group homomorphism
i) only if dim V<= dim W
ii) only if dim V>= dim W
iii) only if dim V=dim W
iv) is always true
(one option is correct and show it by proof)

Explanation / Answer

Let V and W be vector spaces and T:V----->W a linear transformation. Then T(v1 + v2 ) = T(v1) + T(V2) and T(aV1) = a * T(V1) should always be true for any a, V1 and V2. Now we should recall the vector space axioms: A1: (u+v)+w=u+(v+w), associativity A2: there is a vector in V, denoted by 0, such that for any u in V u+0=0+u, additive identity. A3:for each u in V, there is a vector in V, denoted by -u, such that u+(-u)=(-u)+u=0. additive inverse. A4: u+v=v+u, commutativity M1: k(u+v)=ku+kv, for a scalar k M2: (a+b)u=au+bu, for scalars a, b M3: (ab)u=a(bu) M4:1u=u, for unit scalar 1. Now the point of this is to single out the axioms A1-4 because those are the same axioms for a group. In fact, they make a commutative group. In standard Algebra lingo, this is called an abelian group. It is a standard theorem in linear algebra that if T is a linear transformation, then the dimension of V = dimension of W. Now looking back at the definition of linear transformation, it is the definition of a group of homomorphism from V to W. So both V and W are commutative groups.

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