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 ,
We can rewrite this as:
Imagine we are at a party, and is a mapping of boys to girls. The basis for the vector space is the set of traits that collectively “captures” all the boys at the party, so measures how good the boys’ diversity is. measures how good the girls’ diversity is. Thus, captures the inherent difference in diversity of boys to girls. Now, if 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 is doing?
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 is a bad match-maker, it would have a high because it captures a failure of injectivity, i.e. captures how often you’ve mapped two or more boys with the same girl.
On the other hand, measures a success of surjectivity. Essentially, we can think of 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, would’ve likely done a bad job making sure the same girl didn’t have two or more boys mapped to her and also would’ve likely done a better job in making sure all girls had a date.
Adding these two performance metrics of turns out to precisely be equal to the inherent difference diversity of boys to girls.
Back to work.