Bits is a universal currency to transmit information. As long as we can encode

Source-Channel Separation Theorem

“Are we loosing information by using Bits? No.”

Statement:

If a source can be transmitted over a channel at a certain resolution, then it can be transmitted using a binary interface between the source and channel at the same resolution.

Meaning:

If a communication channel has a certain fidelity for arbitrary data, encoding the data by bits will not change the fidelity of the channel.

\begin{align} 0v &= (0+0)v \\ &= 0v+0v \end{align}

Given scalar multiplication is closed, \(0v \in V\), which means \(\exists -0v:0v+(-0v)=0\). Applying that to both sides:

\begin{equation} 0 = 0v\ \blacksquare \end{equation}

The opposite proof of \(\lambda 0=0\) but vectors work the same exact way.

eigenvalue is the scalar needed to scale the basis element of a one dimensional invariant subspace of a Linear Map to represent the behavior of the map:

\begin{equation} Tv = \lambda v \end{equation}

Note we require \(v \neq 0\) because otherwise all scalars count.

eigenvector is a vector that forms the basis list of length 1 of that 1-D invariant subspace under \(T\).

“operators own eigenvalues, eigenvalues own eigenvectors”

Why is eigenvalue consistent per eigenvector? Because a linear map has to act on the same way to something’s basis as it does to the whole space.

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