First Principles AI

Solution

Given:

A=\begin{bmatrix}1&0&1\\2&-1&1\\0&1&2\end{bmatrix}

Message: KEEP IT UP.

Using $A\to 1,\dots,Z\to 26$ and space as $27$ :

K=11,\ E=5,\ E=5,\ P=16,\ \text{space}=27,\ I=9,\ T=20,\ \text{space}=27,\ U=21,\ P=16

Arrange the message into blocks of 3 (column vectors). If needed, pad with space $27$ :

P= \begin{bmatrix} 11&16&20&16\\ 5&27&27&27\\ 5&9&21&27 \end{bmatrix}

Encode by:

C=AP

Compute:

C= \begin{bmatrix}1&0&1\\2&-1&1\\0&1&2\end{bmatrix} \begin{bmatrix} 11&16&20&16\\ 5&27&27&27\\ 5&9&21&27 \end{bmatrix} = \begin{bmatrix} 16&25&41&43\\ 22&14&34&32\\ 15&45&69&81 \end{bmatrix}

\boxed{\text{Encoded message }=\begin{bmatrix}16&25&41&43\\22&14&34&32\\15&45&69&81\end{bmatrix}}