5.1 Linear equations
In practice, we can translate several problems into linear equations. For example, a cash dispenser has $900 in $20 and $50 bills. We know that there are twice as many $20 bills than $50. The question is, how many of each bill does the machine have?
If we denote the number of $20 bills by x1 and the number of $50 bills by x2, we obtain the equations
For two variables, as we have now, these are easily solvable by expressing one in terms of the other. Here, the first equation would imply x1 = 2x2. Plugging it back into the second equation, we obtain 90x2 = 900, which gives x2 = 10. Coming full circle, we can substitute this into x1 = 2x2, yielding the solutions
However, for thousands of variables like in real applications, we need a bit more craft. This is where linear algebra comes in. By introducing the matrix and vectors
the equation can be written in the form Ax = b. That is, in terms of linear transformations, we can reformulate the question: which vector...
