Leave $ B = ((1,2) ^ T, (1,3) ^ T) $ be the base of $ V = Bbb R ^ 2 $.

Find the dual base $ B ^ * = (e_1, e_2) $

Find the matrix of the billinear form: $ T =

e ^ 1 oplus e ^ 2 – e ^ 2 oplus e ^ 1 + 2e ^ 2 oplus e ^ 2 $with respect to.

the canonical basis

Well, I would usually write canonical bases as $ K = ( epsilon ^ 1, epsilon ^ 2) $ and its double base would be $ K ^ * = ( epsilon_1, epsilon_2) = ((1,0) ^ T, (0,1) ^ T) $, I think.

The conditions for dual vectors / covectors are:

$ e_i (e ^ i) = 1 $ Y $ e_i (e ^ j) = 0 $

So we can easily obtain the dual base as $ B ^ * = ((3, -1) ^ T, (- 2,1) ^ T) $ and the condition is met.

But I'm not sure about the $[T]_K $ expression. Probably I would only put the coefficients of that billing form given in the matrix, obtaining a test like this:

$[T]_X = bigl ( begin {smallmatrix} 0 & 1 \ -2 & 1 end {smallmatrix} bigr) $

But it is $ X = B $ or $ X = K $, if we are talking about the bases? Because when I see any problem regarding the canonical form, I would suggest that it is fair $[T]_K $, but we have the main base defined $ B $, and the $ T $ is expressed through $ e ^ i $do not $ epsilon ^ i $.

What if $[T]_B = bigl ( begin {smallmatrix} 0 & 1 \ -2 & 1 end {smallmatrix} bigr) $how could i get $[T]_K $?

My assumption is $[T]_K = (A ^ -1) ^ T[T]_B (A) ^ T $

Is it correct please or I do not understand it at all?