How to compute Basis-transformation matrix

Question

I have the given matrix

{{-1, 4}, {5, -3}}

which is is the simplest representation of the two bases: $$\mathscr{B}=[b_1,b_2]$$ and $$\mathscr{C}=[c_1,c_2]$$

where $b_1=-c_1+4c_2$ and $b_2=5c_1-3c_2$

How do I find the transformation matrix with mathematica for the transformation $\mathscr{B}\rightarrow\mathscr{C}$?

Also how do I find the coordinate vector $[x]_c$ when it is given that $[x]_b=[1,2]$?

Thanks!

Answer 1

m = {{-1, 4}, {5, -3}} ;

Array[b, 2] == m . Array[c, 2] 

Solve[Array[b, 2] == m . Array[c, 2], {b[1], b[2], c[1], c[2]}, Reals] 

Inverse[m] . Array[b, 2] 

Or if you care about nice formatting

Format[c[n_]] := Subscript[c, n];
Format[b[n_]] := Subscript[b, n]; 

MatrixForm[
    m = {{-1, 4}, {5, -3}}
]

TableForm[TraditionalForm/@Thread[ Array[b, 2] == m . Array[c, 2]  ]]

TableForm[TraditionalForm/@Thread[ Array[c, 2] == Inverse[m].Array[b, 2] ]]

TableForm[TraditionalForm/@Thread[ Array[c, 2] == Inverse[m].{1,2} ]]

Answer 2

The matrix: basec2b= {{-1, 4}, {5, -3}} maps the c base vectors to the b base vectors:

baseb == basec2b . basec

A vector: vec with coordinates: coordc in the c base and coordb in b base is:

vec == coordc . basec == coordb . baseb

Now as baseb == basec2b . basec we can write

vec == coordc . basec ==  coordb .  basec2b . basec 

Therefore, as we have basec on the right side in both cases:

coordc == coordb .  basec2b 

Or

coordb == coordc . Inverse[basec2b]

Finally you example with coordb= {1,2}:

coordc == {1,2} . {{-1, 4}, {5, -3}}  == {9,-2}