Answer to Question #283849 in Linear Algebra for DElici

Question #283849

Given two bases



B={1−x,2+x,3−x+x2}



and



C={1,2+x,1+x−x2}



of P2, the vector space of polynomials of degree ≤2,



(i) find p(x)∈P2 whose coordinates with respect to B is [p(x)]B=⎡⎣⎢1 −1 3⎤⎦⎥,



(ii) find the transition (change of coordinates) matrix CMB∈R3×3 from B to C,



(iii) calculate the coordinates [p(x)]C∈R3 of p(x)∈P2 with respect to C.


1
Expert's answer
2022-01-02T16:49:50-0500

Solution:

(i):

The coordinates of p(x)\in P2 with respect to the basis B is [p(x)]B=[113][p(x)]_B=\begin{bmatrix} 1 \\-1\\3\end{bmatrix}

p(x)=1(1x)1(2+x)+3(3x+x2)=(12+9).1+(113)x+3.x2=85x+3x2\therefore p(x)=1(1-x)-1(2+x)+3(3-x+x^2) \\=(1-2+9).1+(-1-1-3)x+3.x^2 \\=8-5x+3x^2

(ii):

Let 1x=a.1+b(2+x)+c(1+xx2)1-x=a.1+b(2+x)+c(1+x-x^2)

1x=(a+2b+c)+(b+c)xcx2\Rightarrow 1-x=(a+2b+c)+(b+c)x-cx^2

On comparing, we get,

a+2b+c=1b+c=1c=0c=0,b=1,a=3a+2b+c=1 \\b+c=-1 \\-c=0 \\\Rightarrow c=0,b=-1,a=3

So, 1x=3(1)1(2+x)+0.(1+xx2) ...(i)1-x=3(1)-1(2+x)+0.(1+x-x^2)\ ...(i)

Also, 2+x=0(1)+1(2+x)+0.(1+xx2) ...(ii)2+x=0(1)+1(2+x)+0.(1+x-x^2)\ ...(ii)

Also, 3x+x2=4(1)+0(2+x)+(1).(1+xx2) ...(iii)3-x+x^2=4(1)+0(2+x)+(-1).(1+x-x^2)\ ...(iii)

From (i), (ii), (iii), we get the transition matrix from B to C as:

CMB=[304110001]_CM_B=\begin{bmatrix} 3&0&4 \\-1&1&0\\0&0&-1\end{bmatrix}

(iii):

From part (i), we have p(x)=85x+3x2p(x)=8-5x+3x^2

Let 85x+3x2=p(1)+q(2+x)+r(1+xx2)8-5x+3x^2=p(1)+q(2+x)+r(1+x-x^2)

85x+3x2=(p+2q+r)+(q+r)xrx2\Rightarrow 8-5x+3x^2=(p+2q+r)+(q+r)x-rx^2

On comparing, we get,

p+2q+r=8q+r=5r=3r=3,q=2,p=15p+2q+r=8 \\q+r=-5 \\-r=3 \\\Rightarrow r=-3,q=-2,p=15

So, we get, p(x)=15(1)2(2+x)3(1+xx2)p(x)=15(1)-2(2+x)-3(1+x-x^2)

So, the coordinates of p(x) w.r.t basis C is:

[p(x)]C=[1523][p(x)]_C=\begin{bmatrix} 15 \\-2\\-3\end{bmatrix}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment