A product of vector spaces is a vector space formed by putting an element from each space into an element of the vector.

constituents

Suppose \(V_1 \dots V_{m}\) are vector spaces over the same field \(\mathbb{F}\)

requirements

Product between \(V_1 \dots V_{m}\) is defined:

\begin{equation} V_1 \times \dots \times V_{m} = \{(v_1, \dots, v_{m}): v_1 \in V_1 \dots v_{m} \in V_{m}\} \end{equation}

“chain an element from each space into another vector”

additional information

operations on Product of Vector Spaces

The operations on the product of vector spaces are defined in the usual way.

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.

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(finished) Project80: Podcast

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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 \(v +\left\{\mqty(a \\ 0): a \in \mathbb{R} \right\}\)).