# A Linear Algebra Analogy

I got destroyed by my LinAlg midterm – didn’t study. On the positive note, here’s a new way I discovered that helped me understand the Rank-Nullity Theorem, which states for a linear transformation $T: V \rightarrow W$,

$\text{dim}(\text{Ran}(T)) + \text{dim}(N(T)) = \text{dim}(V).$

We can rewrite this as:

$\text{dim}(N(T)) + [\text{dim}(\text{Ran}(T)) - \text{dim}(W)]= \text{dim}(V) - \text{dim}(W).$

Imagine we are at a party, and $T$ is a mapping of boys to girls. The basis for the vector space $V$ is the set of traits that collectively “captures” all the boys at the party, so $\text{dim}(V)$ measures how good the boys’ diversity is. $\text{dim}(V)$ measures how good the girls’ diversity is. Thus, $\text{dim}(V) - \text{dim}(W)$ captures the inherent difference in diversity of boys to girls. Now, if $T$ was a good match-maker, he/she would want a good diversity of couples.

How can we determine how bad of a job match-maker $T$ is doing?

$\text{dim}(N(T))$ captures a failure of injectivity, i.e. how often you’ve mapped two or more boys with the same girl. What a waste of diversity!

If $T$ is a bad match-maker, it would have a high $\text{dim}(N(T))$ because it captures a failure of injectivity, i.e. $\text{dim}(N(T))$ captures how often you’ve mapped two or more boys with the same girl.

On the other hand, $\text{dim}(\text{Ran}(T)) -\text{dim}(W)$ measures a success of surjectivity. Essentially, we can think of $\text{dim}(\text{Ran}(T)) -\text{dim}(W)$ as how good of a job T did in making sure all girls had a date.

So, the rewritten form of Rank-Nullity boils down to:

how bad of a job T did in making sure no two boys mapped to the same girl + how good of a job T did in making sure all girls had a date = diversity of boys – diversity of girls

This makes sense, because if there’s just way more boys than girls, $T$ would’ve likely done a bad job making sure the same girl didn’t have two or more boys mapped to her and $T$ also would’ve likely done a better job in making sure all girls had a date.

Adding these two performance metrics of $T$ turns out to precisely be equal to the inherent difference diversity of boys to girls.

Back to work.