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Thread: Linear Algebra Matrix Representation: 1->3, X->2X-1, X^2->X^2-X-1

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    Linear Algebra Matrix Representation: 1->3, X->2X-1, X^2->X^2-X-1

    Problem 9.1. Let [tex]\mathsf{h}:\, \mathcal{P}_2\, \rightarrow \, \mathcal{P}_2[/tex] be the homomorphism given by

    . . . . .[tex]1\, \mapsto\, 3,\quad \mathcal{\large{x}}\, \mapsto\, 2\mathcal{\large{x}}\, -\, 1,\quad \mathcal{\large{x}}^2\, \mapsto\, \mathcal{\large{x}}^2\, -\, \mathcal{\large{x}}\, -\, 1[/tex]

    (a) Find [tex]\mathrm{A}\, =\, \mbox{Rep}_{\mathcal{A,A}}(\mathsf{h})[/tex] where [tex]\mathcal{A}\, =\, \langle 1,\, \mathcal{\large{x}},\, \mathcal{\large{x}}^2 \rangle[/tex]

    (b) Find [tex]\mathrm{B}\, =\, \mbox{Rep}_{\mathcal{B,B}}(\mathsf{h})[/tex] where [tex]\mathcal{B}\, =\, \langle 1,\, 1\, +\, \mathcal{\large{x}},\, 1\, +\, \mathcal{\large{x}}\, +\, \mathcal{\large{x}}^2 \rangle[/tex]

    (c) Find the matrix [tex]\mathrm{P}[/tex] such that [tex]\mathrm{B}\, =\, \mathrm{PAP}^{-1}[/tex]




    Any help on these problems would be awesome! or at least point me in the right direction. Thanks for any help!
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    Last edited by stapel; 12-07-2017 at 07:13 PM. Reason: Typing out the text in the graphic.

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    Elite Member stapel's Avatar
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    Quote Originally Posted by nhallowell View Post
    Problem 9.1. Let [tex]\mathsf{h}:\, \mathcal{P}_2\, \rightarrow \, \mathcal{P}_2[/tex] be the homomorphism given by

    . . . . .[tex]1\, \mapsto\, 3,\quad \mathcal{\large{x}}\, \mapsto\, 2\mathcal{\large{x}}\, -\, 1,\quad \mathcal{\large{x}}^2\, \mapsto\, \mathcal{\large{x}}^2\, -\, \mathcal{\large{x}}\, -\, 1[/tex]

    (a) Find [tex]\mathrm{A}\, =\, \mbox{Rep}_{\mathcal{A,A}}(\mathsf{h})[/tex] where [tex]\mathcal{A}\, =\, \langle 1,\, \mathcal{\large{x}},\, \mathcal{\large{x}}^2 \rangle[/tex]

    (b) Find [tex]\mathrm{B}\, =\, \mbox{Rep}_{\mathcal{B,B}}(\mathsf{h})[/tex] where [tex]\mathcal{B}\, =\, \langle 1,\, 1\, +\, \mathcal{\large{x}},\, 1\, +\, \mathcal{\large{x}}\, +\, \mathcal{\large{x}}^2 \rangle[/tex]

    (c) Find the matrix [tex]\mathrm{P}[/tex] such that [tex]\mathrm{B}\, =\, \mathrm{PAP}^{-1}[/tex]




    Any help on these problems would be awesome! or at least point me in the right direction. Thanks for any help!
    What are your thoughts? What have you tried? How far have you gotten? Where are you stuck? Or are you needing lesson instruction first?

    Thank you!

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