Question about a change of basis algorithm

I have two bases for $$\Bbb{R^2}$$, $$C:=\{(2,-1)^T,(6,-2)^T\}$$ and $$B:=\{(-6,-1)^T,(2,0)^T\}$$. To find the change of basis matrix $$P_{B\to C}$$ we row reduce the system $$\begin{bmatrix}2&6&-6&2 \\-1&-2&-1&0\end{bmatrix}$$

until we have $$\begin{bmatrix}1&0&9&-2 \\0&1&-4&1\end{bmatrix}$$

which gives us the coordinates of the basis vectors of $$B$$ with respect to basis $$C$$ on the columns of the rightwise $$2\times 2$$ matrix, i.e $$[b_1]_C$$ and $$[b_2]_C$$ - these are the columns of the change of basis matrix $$P_{B\to C}$$. I understand some of the connections here, the basis vectors of $$C$$ are just linear combinations of the natural basis of $$\Bbb{R^2}$$ - call it $$E$$. Thus the matrices of $$C$$ and $$E$$ are row equivalent. Why does the same sequence of row operations change the coordinates of the basis vectors of $$B$$ into $$[b_1]_C$$ and $$[b_2]_C$$?

• What is it that you don’t understand? How row-reduction produces $C^{-1}B$ or what that is the correct change-of-basis matrix? – amd Mar 16 '20 at 0:52
• I can't really grasp the theory behind this algorithm/process of finding the change of basis matrix, it seems analogous to the process of finding the inverse of a matrix - but not quite. I'm aware that it is a pretty vague question. – variations Mar 16 '20 at 7:38

Taking the first one first, recall the definition of the coordinates of a vector $$\mathbf v$$ relative to some ordered basis $$\mathcal B=\{\mathbf b_i\}$$: they are the coefficients $$a_i$$ of the basis vectors in the unique linear combination $$\mathbf v = a_1\mathbf b_1+\cdots+a_n\mathbf b_n$$. We generally collect these coefficients into an $$n$$-tuple of scalars that your text denotes by $$[\mathbf v]_{\mathcal B} = a_1[\mathbf b_1]_{\mathcal B}+\cdots+a_n[\mathbf b_n]_{\mathcal B}\in\mathbb F^n$$, where $$\mathbb F$$ is the field over which the vector space is defined. I’ll call this a $$\mathcal B$$-tuple for brevity.
Now let $$M = \begin{bmatrix}[\mathbf b_1]_{\mathcal C}&\cdots&[\mathbf b_n]_{\mathcal C}\end{bmatrix},$$ that is, the matrix with columns equal to the coordinate tuples of the elements of $$\mathcal B$$ relative to some other basis $$\mathcal C$$. Since $$[\mathbf b_j]_{\mathcal B}$$ is just the $$j$$th column of the identity matrix, we have $$M[\mathbf v]_{\mathcal B} = a_1[\mathbf b_1]_{\mathcal C}+\cdots+a_n[\mathbf b_n]_{\mathcal C}.$$ This is a linear combination of $$\mathcal C$$-tuples, so is itself a $$\mathcal C$$-tuple, namely, $$[\mathbf v]_{\mathcal C}$$. Thus, $$M=P_{\mathcal B\to\mathcal C}$$. Since $$M^{-1}M=I$$, it should also be clear that $$M^{-1}$$ maps $$[\mathbf b_j]_{\mathcal C}$$ to $$[\mathbf b_j]_{\mathcal B}$$, so $$P_{\mathcal C\to\mathcal B} = M^{-1}$$.
We can also perform this change of basis in two steps, by first mapping to the standard basis, i.e., $$P_{\mathcal B\to\mathcal C} = P_{\mathcal E\to\mathcal C}P_{\mathcal B\to\mathcal E} = \begin{bmatrix}[\mathbf c_1]_{\mathcal E} & \cdots & [\mathbf c_n]_{\mathcal E}\end{bmatrix}^{-1} \begin{bmatrix}[\mathbf b_1]_{\mathcal E} & \cdots [\mathbf b_n]_{\mathcal E}\end{bmatrix}.$$ In your case, this is $$C^{-1}B$$, with $$B=\begin{bmatrix}-6&2\\-1&0\end{bmatrix}, C=\begin{bmatrix}2&6\\-1&-2\end{bmatrix}.$$
As to the second question regarding computing $$C^{-1}B$$ via row-reduction, remember that every elementary row operation corresponds to left-multiplication by a particular invertible matrix, and so the entire process of row-reduction is equivalent to left-multiplication by some invertible matrix $$E$$. If the matrix $$C$$ is invertible, its RREF is the identity matrix, i.e., $$EC=I$$, from which we have $$E=C^{-1}$$. Because of the way matrix multiplication works, if we augment $$C$$ and reduce it to its RREF, then whatever is on the right side also gets multiplied by $$C^{-1}$$: $$\left[C\mid B\right] \to C^{-1}\left[C\mid B\right] = \left[I\mid C^{-1}B\right],$$ which is exactly what was needed for $$P_{\mathcal B\to\mathcal C}$$. Comparing this to your specific case, the reduced augmented matrix is $$\left[\begin{array}{cc|cc}1&0 & 9&-2 \\ 0&1 & -4&1\end{array}\right]$$ so $$P_{\mathcal B\to\mathcal C}$$ is the submatrix on the right side.
Note that matrix inversion is a special case of this method in which we augment with the identity matrix: $$\left[C\mid I\right] \to C^{-1}\left[C\mid I\right] = \left[I\mid C^{-1}\right].$$