Let \(U_1, \dots, U_{m}\) be subspaces of \(V\); we define a linear

We define \(\Gamma\) to be a map \(U_1 \times \dots U_{m} \to U_1 + \dots + U_{m}\) such that:

\begin{equation} \Gamma (u_1, \dots, u_{m}) = u_1 + \dots + u_{m} \end{equation}

Essentially, \(\Gamma\) is the sum operation of the elements of the tuple made by the Product of Vector Spaces.

\(U_1 + \dots + U_{m}\) is a direct sum IFF \(\Gamma\) is injective

Proof:

This is a work-in-progress page listing all of my production projects.

Fireside: Blog

Fireside Index

20MinuteRants: Blog

https://medium.com/20minuterants

(finished) Project80: Podcast

See .

(finished) Yappin: Podcast

https://anchor.fm/yappin/

So you wanted to be productive?

Go do stuff. Stop reading. Get crap done.

… … …

Wait, you are still here? Well, given that you are sticking around, we might as well discuss some tooling that may help you in organizing your work. By all means I don’t think this is a complete list, many of these have intersecting features; think of this as more a survey of the field—

We mentioned this in class, and I figured we should write it down.

So, if you think about the Product of Vector Space:

\begin{equation} \mathbb{R} \times \mathbb{R} \end{equation}

you are essentially taking the \(x\) axis straight line and “duplicating” it along the \(y\) axis.

Now, the opposite of this is the quotient space:

\begin{equation} \mathbb{R}^{2} / \left\{\mqty(a \\ 0): a \in \mathbb{R} \right\} \end{equation}

Where, we are essentially taking the line in the \(x\) axis and squish it down, leaving us only the \(y\) component freedom to play with (as each element is \(v +\left\{\mqty(a \\ 0): a \in \mathbb{R} \right\}\)).