Prove $\dim(\operatorname{Im}T)=\dim(\operatorname{Im}T')$ Let $T$ be a linear transformation $T:V \rightarrow W$ Of finite-dimensional vector spaces. Is there an easy way to see that prove $\dim(\operatorname{Im}T)=\dim(\operatorname{Im}T')$ for $T'$ being the dual map defined by $T':W' \rightarrow V'$, $f \rightarrow f \circ T$? I am not been able to see why this is true.
 A: Lax has a very insightful proof in his book Linear Algebra and Its Applications. The “four subspaces theorem” (which is beloved by Gilbert Strang) can be generalized to a general finite dimensional vector space using annihilators as a substitute for orthogonal complements. The range of $T’$ is the annihilator of the null space of $T$. So $\text{dim } N(T) + \text{dim } R(T’) = n$. But we also know that $\text{dim } N(T) + \text{dim } R(T) = n$. It follows that $\text{dim } R(T) = \text{dim } R(T’)$.
A: At heart:

Lemma: For $k=\dim(\operatorname{Im}T),$ there exists a basis $w_1,\dots,w_m$ of $W$ and a basis Let $v_1,\dots,v_n$ of $V$ such that $Tv_i=w_i$ for $1\leq i\leq k$ and $Tv_i=0$ for $i>k.$

This is relatively easy to prove. See the outline below.
Then if $w_i’\in W’$ is defined as $w_i’(w_j)=\delta_{ij},$ then $$(T’w_i’)(v_j)=w_i’(Tv_j)=\begin{cases}0&j>k\\\delta_{ij}&j\leq k\end{cases}$$
So $T’(w_i’)=v_i’$ if $i\leq k$ and $T’(w_i’)=0$ if $i>k.$
From here, we can quickly show that $\dim\operatorname{Im}(T’)=k.$ Basically, the image of $T’$ is generated by $v_1’,\dots,v_k’.$

Outline of how to construct the bases of the Lemma.

*

*Pick $w_1,\dots,w_k$ as a basis of $\operatorname{Im}T.$

*Extend to a full basis $w_1,\dots,w_m$ of $W.$

*Pick any $v_1,\dots,v_k$ such that $Tv_i=w_i.$ Show these are linearly independent.

*Show you can extend $v_i$ to basis $v_1,\dots,v_k,v_{k+1},\dots,v_n$ with $Tv_i=0$ for $i>k.$
That last step is the slightly tricky part. But if $Tv_j\neq0$ for $j>k,$ we can write $Tv_j=\sum_{i=1}^k a_iw_i$ for scalars $a_i$ then replace $v_j$ with:
$$u_j=v_j-\sum_{i=1}^ka_iv_i$$ and still have a basis. And $Tu_j=0.$
