Check the solution to "Let A and C be the following ordered..."

College
Algebra
Linear algebra
Alternate coordinate systems

ringearV

Answered question

2021-02-14

Let A and C be the following ordered bases of P3(t):

A = (1, 1 + t, 1 + t + t^{2}, 1 + t + t^{2} + t^{3})

C = (1, - t, t^{2} - t^{3})

Find tha change of coordinate matrix

I_{C A}

Answer & Explanation

irwchh

Skilled2021-02-15Added 102 answers

Solution: $A = (1, 1 + t, 1 + t + t^{2}, 1 + t + t^{2} + t^{3}) and C = (1, - t, t^{2}, - t^{3})$ To find the chance of coordinate matrix $I_{C A}$ :
$1 = a_{1} (1) + a_{2} (1 + t) + a_{3} (1 + t + t^{2}) + a_{4} (1 + t + T^{2} + t^{3})$
$\Rightarrow a_{1} + a_{2} + a_{3} + a_{4} = 1, a_{2} + a_{3} + a_{4} = 0, a_{3} + a_{4} = 0 and a_{4} = 0$
$\Rightarrow a_{1} = 1$
$- t = b_{1} (1) + b_{2} (1 + t) + b_{3} (1 + t + t_{2}) + b_{4} (1 + t + t^{2} + t^{3})$
$R i > \leftrightarrow o w b_{1} + b_{2} + b_{3} + b_{4} = 0, b_{2} + b_{3} + b_{4} = - 1, b_{3} + b_{4} = 0 and b_{4} = 0$
$\Rightarrow b_{1} = 1 and b_{2} = - 1$
$t^{2} = c_{1} (1) + c_{2} (1 + t) + c_{3} (1 + t + t^{2}) + c_{4} (1 + t + t^{2} + t^{3})$
$\Rightarrow c_{1} + c_{2} + c_{3} + c_{4} = 0, c_{2} + c_{3} + c_{4} = 0, c_{3} + c_{4} = 1 and c_{4} = 0$
$\Rightarrow c_{2} = - 1, c_{3} = 1$

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