I'm currently learning eigenvectors and Linear Transformations. And this basis change part is particularly confusing for me..
I feel like there are two types of questions,
- First type, you have a matrix with basis $B =\{b_1,b_2\}$ and then you change it to basis $C = \{c_1,c_2,c_3\}$.
ex) $T(b_1) = 3c_1-2c_2+5c_3$ and $T(b_2) = 4c_1+7c_2-c_3$.
In this case the coefficients of the $c_1,c_2,c_3$ become the column vectors for the transformation matrix, so $\pmatrix{3&4\cr -2&7\cr 5&-1\cr}$. (NOTE: original was [(3,-2,5),(4,7,-1)] but the intent was clear from context.)
- Second type, you are given $T\pmatrix{x_1\cr x_2\cr} = \pmatrix{x_1+3x_2\cr 5x_2-x_1\cr 4x_1+x_2\cr}$. Then the transformation matrix is transformed into a matrix whose first column vector is coefficients of $x_1$ and that of $x_2$ in the second column, so $\pmatrix{1&3\cr -1&5\cr4&1\cr}$. (NOTE: Original was [(1,-1,4),(3,5,1)].)
What's the difference between these two, and how come there are differences in getting the transfomration matrix between the two? Aren't they both just basis change?