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Math Help - creating a matrix

  1. #1
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    creating a matrix

    Hi I need to know how to create a matrix [T]_{B}
    where V=P_{2}(\mathbb{R}) is the vectorspace and
    T(f(x))=f(0) +f(1)(x+x^2)
    B={1,x,x^2}, the standard basis.
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    Hi I need to know how to create a matrix [T]_{B}
    where V=P_{2}(\mathbb{R}) is the vectorspace and
    T(f(x))=f(0) +f(1)(x+x^2)
    B={1,x,x^2}, the standard basis.
    There is a standard technique for constructing the matrix corresponding to a given linear transformation in a given basis:

    1) Apply the linear transformation to each basis member in turn.
    2) Write the result as a linear combination of the basis members.
    3) The coefficients in that linear combination form the columns of the matrix.

    For example, the first basis "vector" is the constant function 1: f(x)= 1 so f(0)= 1 and f(1)= 1. T(1)= 1+ 1(x+ x^2= 1+ 1x+ 1x^2. The coefficients are "1 1 1" so the first column of the matrix is \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}.

    The next basis "vector" is f(x)= x. f(0)= 0 and f(1)= 1. What is T(x)? What is the second column of the matrix?
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