In the following example, it is meant to illustrate how to solve change of basis problems, however, I am having trouble following this example as it seems to be missing some intermediate steps. I understand how to solve part $a$, but the solution to part $b$ is where I am unable to follow.
"A short calculation shows that $[(1,2)]_C = ...$ "
The matrix contains -3 and 5.5, however, at no point does the text illustrate how this was done. An explanation for obtaining the first matrix containing the -3 and 5.5 would suffice (since the others follow the same general principle).
(For self-studying a Linear Algebra course over the summer, any help would be appreciated!)
The idea is to express the basis vectors $(1,2)$ and $(3,4)$ whose components are given with respect to the canonical basis $\{ (1,0) , (0,1) \}$ by the vectors of the basis $C$. So what we need to solve is the problem: "how do we express $(1,2)$ as a linear combination of the basis vectors?", i.e. "how many times do we have to add basis vector one and basis vector two to get $(1,2$)" which translates to the equation $$ (1,2) = \alpha \cdot (7,3) + \beta \cdot (4,2). $$ We can write this as a matrix equation for the unknown vector $(\alpha,\beta)$ (describing the scaling factors for the two basis vectors of $C$) as follows: $$ \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 7 &4 \\ 3 &2 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} =: A x. $$ With the matrix $A$ and the unknown vector $x = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$. This linear system of equations can then be solved either using the Gauss algorithm or using the neat formula to calculate the inverse matrix of a 2x2 matrix $$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ by $$ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d &-b \\ -c &a \end{pmatrix} $$ and then $x = A^{-1} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} -3 \\ 5.5\end{pmatrix}$.