Find out the answer to "Let γ={t2−t+1,t+1,t2+1}andβ={t2+t+4,4t2−3t+2,2t2+3}beorderedbasesforP2(R). Find the change of coordinate matrix..."

College
Algebra
Linear algebra
Alternate coordinate systems

e1s2kat26

Answered question

2020-11-01

Let $γ = {t^{2} - t + 1, t + 1, t^{2} + 1} and β = {t^{2} + t + 4, 4 t^{2} - 3 t + 2, 2 t^{2} + 3} b e or d e r e d b a s e s f or P_{2} (R) .$ Find the change of coordinate matrix Q that changes $β coordinates into γ - coordinates$

Answer & Explanation

i1ziZ

Skilled2020-11-02Added 92 answers

Let $t^{2} + t + 4 = γ_{1} (t^{2} - t + 1) + γ_{2} (t = 1) + γ_{3} (t^{2} + 1)$
$\Rightarrow γ_{1} + γ_{3} = 1, - γ_{1} + γ_{2} = 1, γ_{1} + γ_{2} + γ_{3} = 4$
$\Rightarrow 1 + γ_{2} = 4$
$\Rightarrow γ_{2} = 3$
$- γ_{1} + γ_{2} = 1 \Rightarrow - γ_{1} + 3 = 1 \Rightarrow γ_{1} = 2$
$γ_{1} + γ_{3} = 1 \Rightarrow 2 + γ_{3} = 1 \Rightarrow γ_{3} = - 1$ Let $4 t^{2} - 3 r + 2 = γ_{1} (t^{2} - t + 1) + γ_{2} (t + 1) + γ_{3} (t^{2} + 1)$
$\Rightarrow γ_{1} + γ_{3} = 4, - γ_{1} + γ_{2} = - 3, γ_{1} + γ_{2} + γ_{3} = 2$
$\Rightarrow 4 + γ_{2} = 2$
$\Rightarrow γ_{2} = - 2$
$- γ_{1} + γ_{2} = - 3 \Rightarrow - γ_{1} - 2 = - 3 \Rightarrow γ_{1} = 1$
$γ_{1} + γ_{3} = 4 \Rightarrow 1 + γ_{3} = 4 \Rightarrow γ_{3} = 3$ Let $2 t^{2} + 3 = γ_{1} (t^{2} - t + 1) + γ_{2} (t + 1) + γ_{3} (t^{2} + 1)$
$\Rightarrow γ_{1} + γ_{3} = 2, - γ_{1} + γ_{2} = 0, γ_{1} + γ_{2} + γ_{3} = 3$
$\Rightarrow 2 + γ_{2} = 3$
$\Rightarrow γ_{2} = 1$
$- γ_{1} + γ_{2} = 0 \Rightarrow - γ_{1} + 1 = 0 R i > \leftrightarrow o w γ_{1} = 1$
$γ_{1} + γ_{3} = 2 \Rightarrow 1 + γ_{3} = 2 \Rightarrow γ_{3} = 1$

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