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

nhallowell

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Problem 9.1. Let \(\displaystyle \mathsf{h}:\, \mathcal{P}_2\, \rightarrow \, \mathcal{P}_2\) be the homomorphism given by

. . . . .\(\displaystyle 1\, \mapsto\, 3,\quad \mathcal{\large{x}}\, \mapsto\, 2\mathcal{\large{x}}\, -\, 1,\quad \mathcal{\large{x}}^2\, \mapsto\, \mathcal{\large{x}}^2\, -\, \mathcal{\large{x}}\, -\, 1\)

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

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

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




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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stapel

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Problem 9.1. Let \(\displaystyle \mathsf{h}:\, \mathcal{P}_2\, \rightarrow \, \mathcal{P}_2\) be the homomorphism given by

. . . . .\(\displaystyle 1\, \mapsto\, 3,\quad \mathcal{\large{x}}\, \mapsto\, 2\mathcal{\large{x}}\, -\, 1,\quad \mathcal{\large{x}}^2\, \mapsto\, \mathcal{\large{x}}^2\, -\, \mathcal{\large{x}}\, -\, 1\)

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

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

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




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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