I'm having problems finding the Matrix T of the following linear transformation
$T\::\:R2\left(x\right)\:->R2\left(x\right)\:defined\:by\:T\left(ax^2+bx+c\right)=2ax+b$
so I know that a base of T is $(x^2,x,1)$, but im not able to get the im(T) to make the matrix any help?
Polynomials are functions that are defined by the numbers that multiply $1, x, x ^ 2, x ^ 3, ....$ in the plots of the polynomial. That is, a polynomial of degree 2 or less, $ax ^ 2 + bx + c$, is defined by the numbers (or "coefficients") $(a, b, c)$.
If the base of ${R} _2 [x]$ (vector space of polynomials of degree not greater than 2) is $(x ^ 2, x, 1)$, the matrix of the linear application will have in the first, second and third columns, respectively, the coefficients of $T (x ^ 2)$, $T (x)$ and $T (1)$, that is, the coefficients of
$2x = 0 * x ^ 2 + 2 * x + 0 * 1$ (first column is $[0 \ 2 \ 0] ^ T$,
$1 = 0 * x ^ 2 + 0 * x + 1 * 1$ (second column is $[0 \ 0 \ 1] ^ T$ and
$0 = 0 * x ^ 2 + 0 * x + 0 * 1$ (third column is $[0 \ 0 \ 0] ^ T$.
as auxiliary calculus we have,
canonical basis $(x ^ 2, x, 1)$
$T (x ^ 2) = T (1x ^ 2 + 0x + 0) = 2 * 1 * x + 0 = 2x$
$T (x) = T (0x ^ 2 + x + 0) = 2 * 0x + 1 = 1$
$T (1) = T (0x ^ 2 + 0x + 1) = 0$
solution $(2x, 1,0)$