- #1

Adgorn

- 130

- 18

## Homework Statement

Suppose T:V→U is linear and V has finite dimension. Prove that I am T

^{t}= (Ker T)

^{0}

## Homework Equations

dim(W)+dim(W

^{0})=dim(V) where W is a subspace of V and V has finite dimension.

## The Attempt at a Solution

I first proved I am T

^{t}⊆ (Ker T)

^{0}. Let u be an arbitrary element of Ker T and T

^{t}(Φ )∈ I am T

^{t}, then T

^{t}(Φ)(u)=Φ(T(u))=Φ(0)=0. Thus, since all elements of I am T

^{t}map all elements of Ker T into 0, I am T

^{t}⊆ (Ker T)

^{0}.

I now only need to proof dim((Ker T)

^{0}) = dim I am T

^{t}. Since V has finite dimension, dim((Ker T)

^{0}) = dim(V)-dim(Ker T), but dim(Im T) = dim(V)-dim(Ker T). Thus, dim((Ker T)

^{0})=dim(Im T). This is where I got stuck, how do I prove dim (Im T) = dim ((Im T)

^{0})?