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

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nhallowell

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Problem 9.1. Let h: P2 → P2\displaystyle \mathsf{h}:\, \mathcal{P}_2\, \rightarrow \, \mathcal{P}_2 be the homomorphism given by

. . . . .1 ↦ 3,x ↦ 2x − 1,x2 ↦ x2 − x − 1\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 A = ⟨1, x, x2⟩\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 B = ⟨1, 1 + x, 1 + x + x2⟩\displaystyle \mathcal{B}\, =\, \langle 1,\, 1\, +\, \mathcal{\large{x}},\, 1\, +\, \mathcal{\large{x}}\, +\, \mathcal{\large{x}}^2 \rangle

(c) Find the matrix P\displaystyle \mathrm{P} such that B = PAP−1\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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nhallowell said:
Problem 9.1. Let h: P2 → P2\displaystyle \mathsf{h}:\, \mathcal{P}_2\, \rightarrow \, \mathcal{P}_2 be the homomorphism given by

. . . . .1 ↦ 3,x ↦ 2x − 1,x2 ↦ x2 − x − 1\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 A = ⟨1, x, x2⟩\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 B = ⟨1, 1 + x, 1 + x + x2⟩\displaystyle \mathcal{B}\, =\, \langle 1,\, 1\, +\, \mathcal{\large{x}},\, 1\, +\, \mathcal{\large{x}}\, +\, \mathcal{\large{x}}^2 \rangle

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