how to do this compute the matrix of $$ with respect to the basis $B={1,x,x^2}$












0












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Let $V = mathbb{R}[x]_{le2}$ be the $mathbb{R}-$vector space of polynomials
$f(x) = a_0 + a_1x + a_2x^2$ with real coefficients $a_i$ in $mathbb{R}$, of degree $le2$.



Define,
$$langle f(x),g(x)rangle =int_{-2}^{2}f(x)g(x)dx$$



Compute the matrix of $langle .,.rangle$ with respect to the basis $B = {1, x, x^2}$






linear-algebra







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    Welcome to Math.SE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
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    – platty
    Nov 30 '18 at 2:29
















0












$begingroup$


Let $V = mathbb{R}[x]_{le2}$ be the $mathbb{R}-$vector space of polynomials
$f(x) = a_0 + a_1x + a_2x^2$ with real coefficients $a_i$ in $mathbb{R}$, of degree $le2$.



Define,
$$langle f(x),g(x)rangle =int_{-2}^{2}f(x)g(x)dx$$



Compute the matrix of $langle .,.rangle$ with respect to the basis $B = {1, x, x^2}$






linear-algebra







share|cite|improve this question











$endgroup$












  • $begingroup$
    Welcome to Math.SE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    $endgroup$
    – platty
    Nov 30 '18 at 2:29














0












0








0





$begingroup$


Let $V = mathbb{R}[x]_{le2}$ be the $mathbb{R}-$vector space of polynomials
$f(x) = a_0 + a_1x + a_2x^2$ with real coefficients $a_i$ in $mathbb{R}$, of degree $le2$.



Define,
$$langle f(x),g(x)rangle =int_{-2}^{2}f(x)g(x)dx$$



Compute the matrix of $langle .,.rangle$ with respect to the basis $B = {1, x, x^2}$






linear-algebra







share|cite|improve this question











$endgroup$




Let $V = mathbb{R}[x]_{le2}$ be the $mathbb{R}-$vector space of polynomials
$f(x) = a_0 + a_1x + a_2x^2$ with real coefficients $a_i$ in $mathbb{R}$, of degree $le2$.



Define,
$$langle f(x),g(x)rangle =int_{-2}^{2}f(x)g(x)dx$$



Compute the matrix of $langle .,.rangle$ with respect to the basis $B = {1, x, x^2}$






linear-algebra




linear-algebra






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Nov 30 '18 at 4:44









Sujit Bhattacharyya

1,131419




1,131419










asked Nov 30 '18 at 2:24









vickyvicky

1




1












  • $begingroup$
    Welcome to Math.SE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    $endgroup$
    – platty
    Nov 30 '18 at 2:29


















  • $begingroup$
    Welcome to Math.SE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    $endgroup$
    – platty
    Nov 30 '18 at 2:29
















$begingroup$
Welcome to Math.SE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– platty
Nov 30 '18 at 2:29




$begingroup$
Welcome to Math.SE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
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– platty
Nov 30 '18 at 2:29










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$begingroup$

Define the variables $$mathbf a = (a_0,a_1,a_2)^T, mathbf b = (b_0,b_1,b_2)^T, text{ and } P = begin{pmatrix}p_{00} & p_{01} & p_{02} \ p_{10} & p_{11} & p_{12} \ p_{20} & p_{21} & p_{22} end{pmatrix}$$ Also, let $$f(x) = a_0+a_1x+a_2x^2 text{ and } g(x) = b_0+b_1x+b_2x^2$$



Then, $P$ is a matrix for $langlecdot , cdotrangle$ iff



$$ mathbf a^T P mathbf b = langle f(x), g(x)rangle = int_{-2}^2 f(x)g(x)text dx$$



Evaluating the left-hand side gives



$$ mathbf a^T P mathbf b = sumlimits_{i=0}^2sumlimits_{j=0}^2 p_{ij}a_ib_j$$



so the $ij$-th entry of $P$ is the coefficient of $a_ib_j$. Evaluating the right-hand side gives



$$ int_{-2}^2 f(x)g(x)text dx = int_{-2}^2 (a_0+a_1x+a_2x^2)(b_0+b_1x+b_2x^2)text dx \ = 4a_0b_0 + frac{16}{3}a_0b_2 + frac{16}{3}a_1b_1 + frac{16}{3}a_2b_0 + frac{64}{5}a_2b_2$$



Putting this information together, we can deduce the entries of the matrix $P$:



$$ P = begin{pmatrix}4 & 0 & frac{16}{3} \ 0 & frac{16}{3} & 0 \ frac{16}{3} & 0 & frac{64}{5} end{pmatrix} $$






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    $begingroup$

    Define the variables $$mathbf a = (a_0,a_1,a_2)^T, mathbf b = (b_0,b_1,b_2)^T, text{ and } P = begin{pmatrix}p_{00} & p_{01} & p_{02} \ p_{10} & p_{11} & p_{12} \ p_{20} & p_{21} & p_{22} end{pmatrix}$$ Also, let $$f(x) = a_0+a_1x+a_2x^2 text{ and } g(x) = b_0+b_1x+b_2x^2$$



    Then, $P$ is a matrix for $langlecdot , cdotrangle$ iff



    $$ mathbf a^T P mathbf b = langle f(x), g(x)rangle = int_{-2}^2 f(x)g(x)text dx$$



    Evaluating the left-hand side gives



    $$ mathbf a^T P mathbf b = sumlimits_{i=0}^2sumlimits_{j=0}^2 p_{ij}a_ib_j$$



    so the $ij$-th entry of $P$ is the coefficient of $a_ib_j$. Evaluating the right-hand side gives



    $$ int_{-2}^2 f(x)g(x)text dx = int_{-2}^2 (a_0+a_1x+a_2x^2)(b_0+b_1x+b_2x^2)text dx \ = 4a_0b_0 + frac{16}{3}a_0b_2 + frac{16}{3}a_1b_1 + frac{16}{3}a_2b_0 + frac{64}{5}a_2b_2$$



    Putting this information together, we can deduce the entries of the matrix $P$:



    $$ P = begin{pmatrix}4 & 0 & frac{16}{3} \ 0 & frac{16}{3} & 0 \ frac{16}{3} & 0 & frac{64}{5} end{pmatrix} $$






    share|cite|improve this answer









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      0












      $begingroup$

      Define the variables $$mathbf a = (a_0,a_1,a_2)^T, mathbf b = (b_0,b_1,b_2)^T, text{ and } P = begin{pmatrix}p_{00} & p_{01} & p_{02} \ p_{10} & p_{11} & p_{12} \ p_{20} & p_{21} & p_{22} end{pmatrix}$$ Also, let $$f(x) = a_0+a_1x+a_2x^2 text{ and } g(x) = b_0+b_1x+b_2x^2$$



      Then, $P$ is a matrix for $langlecdot , cdotrangle$ iff



      $$ mathbf a^T P mathbf b = langle f(x), g(x)rangle = int_{-2}^2 f(x)g(x)text dx$$



      Evaluating the left-hand side gives



      $$ mathbf a^T P mathbf b = sumlimits_{i=0}^2sumlimits_{j=0}^2 p_{ij}a_ib_j$$



      so the $ij$-th entry of $P$ is the coefficient of $a_ib_j$. Evaluating the right-hand side gives



      $$ int_{-2}^2 f(x)g(x)text dx = int_{-2}^2 (a_0+a_1x+a_2x^2)(b_0+b_1x+b_2x^2)text dx \ = 4a_0b_0 + frac{16}{3}a_0b_2 + frac{16}{3}a_1b_1 + frac{16}{3}a_2b_0 + frac{64}{5}a_2b_2$$



      Putting this information together, we can deduce the entries of the matrix $P$:



      $$ P = begin{pmatrix}4 & 0 & frac{16}{3} \ 0 & frac{16}{3} & 0 \ frac{16}{3} & 0 & frac{64}{5} end{pmatrix} $$






      share|cite|improve this answer









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        0








        0





        $begingroup$

        Define the variables $$mathbf a = (a_0,a_1,a_2)^T, mathbf b = (b_0,b_1,b_2)^T, text{ and } P = begin{pmatrix}p_{00} & p_{01} & p_{02} \ p_{10} & p_{11} & p_{12} \ p_{20} & p_{21} & p_{22} end{pmatrix}$$ Also, let $$f(x) = a_0+a_1x+a_2x^2 text{ and } g(x) = b_0+b_1x+b_2x^2$$



        Then, $P$ is a matrix for $langlecdot , cdotrangle$ iff



        $$ mathbf a^T P mathbf b = langle f(x), g(x)rangle = int_{-2}^2 f(x)g(x)text dx$$



        Evaluating the left-hand side gives



        $$ mathbf a^T P mathbf b = sumlimits_{i=0}^2sumlimits_{j=0}^2 p_{ij}a_ib_j$$



        so the $ij$-th entry of $P$ is the coefficient of $a_ib_j$. Evaluating the right-hand side gives



        $$ int_{-2}^2 f(x)g(x)text dx = int_{-2}^2 (a_0+a_1x+a_2x^2)(b_0+b_1x+b_2x^2)text dx \ = 4a_0b_0 + frac{16}{3}a_0b_2 + frac{16}{3}a_1b_1 + frac{16}{3}a_2b_0 + frac{64}{5}a_2b_2$$



        Putting this information together, we can deduce the entries of the matrix $P$:



        $$ P = begin{pmatrix}4 & 0 & frac{16}{3} \ 0 & frac{16}{3} & 0 \ frac{16}{3} & 0 & frac{64}{5} end{pmatrix} $$






        share|cite|improve this answer









        $endgroup$



        Define the variables $$mathbf a = (a_0,a_1,a_2)^T, mathbf b = (b_0,b_1,b_2)^T, text{ and } P = begin{pmatrix}p_{00} & p_{01} & p_{02} \ p_{10} & p_{11} & p_{12} \ p_{20} & p_{21} & p_{22} end{pmatrix}$$ Also, let $$f(x) = a_0+a_1x+a_2x^2 text{ and } g(x) = b_0+b_1x+b_2x^2$$



        Then, $P$ is a matrix for $langlecdot , cdotrangle$ iff



        $$ mathbf a^T P mathbf b = langle f(x), g(x)rangle = int_{-2}^2 f(x)g(x)text dx$$



        Evaluating the left-hand side gives



        $$ mathbf a^T P mathbf b = sumlimits_{i=0}^2sumlimits_{j=0}^2 p_{ij}a_ib_j$$



        so the $ij$-th entry of $P$ is the coefficient of $a_ib_j$. Evaluating the right-hand side gives



        $$ int_{-2}^2 f(x)g(x)text dx = int_{-2}^2 (a_0+a_1x+a_2x^2)(b_0+b_1x+b_2x^2)text dx \ = 4a_0b_0 + frac{16}{3}a_0b_2 + frac{16}{3}a_1b_1 + frac{16}{3}a_2b_0 + frac{64}{5}a_2b_2$$



        Putting this information together, we can deduce the entries of the matrix $P$:



        $$ P = begin{pmatrix}4 & 0 & frac{16}{3} \ 0 & frac{16}{3} & 0 \ frac{16}{3} & 0 & frac{64}{5} end{pmatrix} $$







        share|cite|improve this answer












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        answered Nov 30 '18 at 5:46









        AlexanderJ93AlexanderJ93

        6,163823




        6,163823






























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