Findthe Matrix T of the following linear transformationT:R2(x−>R2(x)definedbyT(ax^2+bx+c)=2ax+b

nar6jetaime86 2022-09-13 Answered
Findthe Matrix T of the following linear transformation
T : R 2 ( x ) > R 2 ( x ) d e f i n e d b y T ( a x 2 + b x + c ) = 2 a x + b
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Answers (1)

Harper Brewer
Answered 2022-09-14 Author has 16 answers
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, a x 2 + b x + 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
2 x = 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 ( 1 x 2 + 0 x + 0 ) = 2 1 x + 0 = 2 x
T ( x ) = T ( 0 x 2 + x + 0 ) = 2 0 x + 1 = 1
T ( 1 ) = T ( 0 x 2 + 0 x + 1 ) = 0
solution ( 2 x , 1 , 0 )

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