All this week San Diego Convention Center is hosting the Joint Mathematical Meeting that is bringing mathematicians across the country under one roof to discuss mathematics. Of course there are many publishers present and products and services being promoted to the mathematics community too.

Attending the conference has encouraged me to do math again. Here what follows is a little math to begin the formal road to linear algebra.

The first fact states for subspaces \( U_1, \ldots , U_m\) of a vector space \(V\) and some elements \(u_1 \in U_1, \ldots, u_m \in U_m\) that

$$U = U_1 + \ldots + U_m \text{ is a direct sum}$$ $$ \Leftrightarrow$$ $$\text{the only way to express } 0 \in U \text{ as a sum is for } u_1 = \dots = u_m = 0$$

In words, we then have that the sum of subspaces \(U_1 + \ldots + U_m\) is a direct sum if and only if the only one way to express the additive identity \(0\) is for when \(u_1, \ldots u_m = 0\).

Sheldon Axler then follows the above fact with the statement that for subspaces \(U, W\) of the vector space \(V\), $$ U \cap W = \{0\} \Leftrightarrow U + W \text{ is a direct sum}$$

At this point, I'll be continuing my formal journey in algebra with this textbook. Working with the book will help to open new worlds and insights in mathematics.