Problem 63 - College Algebra

Question

It's January \(1,\) and you've written down your major goal for the year. You do not want those closest to you to see what you've written in case you do not accomplish your objective. Consequently, you decide to use a coding matrix to encode your goal. Explain how this can be accomplished.

Step-by-Step Solution

Answer

By encoding the letters of the goal into numbers and using a selected encoding matrix, one can encode the messages into a new set of numbers that represent the written goal. This encoded message can then be decoded using the inverse operation if the encoding matrix is known and it's inverts matrix exists.

Step 1: Understanding Encoding

An encoding scheme can have many different forms, but for simplicity, let's choose to encode each letter with an integer. For example, A would equal 1, B would equal 2, C would equal 3, and so on, up to Z equaling 26. Once that is chosen, next is to form vectors of these numbers from the goal you've written down.

Step 2: Select Encoding Matrix

An encoding matrix is a key component in this encoding scheme. For example, you might select a 2x2 matrix due to its simplicity. Let's say the matrix \(M\) you chose is \(\[ \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix} \]\). This matrix would be used for each pair of integers from your encoded message.

Step 3: Matrix Multiplication

Take the first two integers and form a column vector. This vector is then multiplied by the encoding matrix to a resulting encoded vector. For example, if first two letters of the goal were 'AN' or represented as '1 14' then after the matrix multiplication, it might result in an encoded message vector of '45 72'.

Step 4: Continue the Process

Continue the process for all pairs of integers that represent your written down goal. If there is an odd number of integers, you can pad the last one with a zero or another suitable number stand for space or punctuation. This will result into a series of pairs of new numbers that represent the encoded message.