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    Define a linear transformation. T: P_2 --> R^2 by

    T(p) = \begin{bmatrix} p(0) \\p(0) \end{bmatrix}.

    Find polynomials p_1 and p_2 in P_2 that span the kernel of T, and describe the range of T.

    My answer:

    This is the definition of a linear transformation:

    T(p+q) = \begin{bmatrix} (p+q)(0) \\(p+q)(0) \end{bmatrix}

    = \begin{bmatrix} p(0) \\p(0) \end{bmatrix} + \begin{bmatrix} q(0) \\q(0) \end{bmatrix}

    = T(p) + T(q)

    T(cp) = \begin{bmatrix} (cp)(0) \\(cp)(0) \end{bmatrix}

    = c\begin{bmatrix} p(0) \\p(0) \end{bmatrix}= cT(p)

    These are the polynomials p_1 and p_2 in P_2:

    T(p) = \begin{bmatrix} p(0) \\p(0) \end{bmatrix} = \begin{bmatrix} 0 \\0 \end{bmatrix}

    So p(0) = 0 and p(0) = 0...

    I chose any two polynomials that met these requirments...so...

    p_1 = x
    p_2 = x^2

    But I'm somewhat confused about the range...how can I find it?

    Thank you in advance.
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    You weren't asked to show that T is linear so don't bother.

    Your basis for the kernel is correct but needs more justification. Start with p(x)=a_2x^2+a_1x+a_0 . Then p(0)=a_0 , etc.

    Range is \{\left[ \begin{array}{c}

a_0\\

a_0

\end{array}\right]:a_0\in \mathbb{R}\} , which is spanned by?
 
 
 
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Updated: April 10, 2011

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