Update 2024-07-27-rref.md

_posts/2024-07-27-rref.md

@@ -67,7 +67,7 @@ Now, some smart-asses (many of which, I love) might object to what I'm about to
 
 First, though, I'm going to give you the completely oversimplified low-down of what the majority of abstract algebra is like, on the motivational level.
 
-> "We have some things that interact in some nice way, let's look at functions that preserve those interactions."
+> "We have some things that interact in some nice way, let's look at functions and substructures that preserve those interactions."
 
 In terms of linear algebra, our "things" are vectors (over some field of scalars), and the interaction is vector addition and scalar multiplication. 
 
@@ -81,9 +81,11 @@ So, in a sense, we can view vector spaces as some collection of vectors, and we
 
 NOTE: Saying scalars from a "field" is important! There are other nonvector things we can use to "scale/multiply to" vectors, but being able to divide nonzero scalars is what makes linear algebra so nice (and what makes Modules--vector spaces over a ring instead of a field--much crazier). 
 
-So, algebraists, then, are interested in functions that preserve the interactions of vectors (which are linear combinations in our case): called **Linear Transformations**. 
+So, algebraists, then, are interested in functions and substructures of the vector space that preserve the interactions of vectors (which are linear combinations in our case). The substructures of vector spaces that preserve linear combinations are **subspaces**, and the functions that preserve linear combinations are called **Linear Transformations**. Our focus today is on linear transformations, though.
 
-$$T(v)=T(c_1v_1+\ldots+c_nv_n)=c_1T(v_1)+\ldots+c_nT(v_n)$$
+$$\begin{gather*}
+T(v)=T(c_1v_1+\ldots+c_nv_n)\\=c_1T(v_1)+\ldots+c_nT(v_n)
+\end{gather*}$$
 
 But why are *you* (someone who is probably *not* an algebraist) interested in functions that preserve linear combinations? Well, if you're taken calculus, then you definitely are! How do you take the derivative of a polynomial?