Basis Transformation, or By = x

The student numbers can generally be described as . Which means, In bed are exactly the students whoa are nowhere else. As said, we want to transform this into B by solving By = x.

$$\left( \begin{array}{cccc\vert c}3 & 0 & 3 & 11 & a\\ -18& 0... ... & 4& 9 &c \\ -2& -1 & 1 & 16 & 880-a-b-c \end{array} \right)$$

which leads to the following

$$\left( \begin{array}{cccc\vert c}3 & 0 & 3 & 11 & a\\ -18& ... ... b\\ 17& 1 & 4& 9 & c\\ 0& 0 & 0 & 1 & 22 \end{array} \right)$$

Before we continue solving this matrix problem, we think about what we want to do. We want to calculate the numbers of students after n lectures, which basically means . But is becoming more and more like shown below

$$\mathbf D^n = \left( \begin{array}{cccc}0 & 0 & 0 & 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{array} \right)$$

This means that all we need is , which can be read out as 22 from the last line in the changed version of By = x. We now have to retransfer y back into the standard basis, for those who don't know this means multiplying with 22. And this is the result.

242 students are going in the correct lecture
88 students are going in the wrong lecture
198 students are going in the pub
358 students are staying in bed

As we can see the final numbers of students doesn't depend on the starting values a, b and c. Therefore we have answered the additional question as well, if we only chose n big enough.