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a) Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to 2. Let C={3,1+x, 3−3x+3x^2} be an ordered basis for P2.

b) Write −3+12x−9x^2 as a linear combination of elements from the basis C: −3+12x−9x2= (3)(c1)+ (1+x)(c2) +(3−3x+3x2)(c3).

c) Let [q]C denote the coordinate representation of qq relative to the basis C. Find the coordinate vector representation for −3+12x−9x^2 relative to the basis C. Your answer should be a vector of the general form <1,2,3>.

Respuesta :

ezeebuka06

Answer:

Coordinate vedctor representation of [tex]-3+12x-9x^{2}[/tex] relative to basis C =(C₁ ,C₂, C₃) = ( 1, 2, 3)

Step-by-step explanation:

Let -3+12x-9x^{2}= C₁(3) + C₂(1+x)+C₃(3-3x+[tex]3x^{2}[/tex])=

3C₁ +C₂ +3C₂=-3-------------(1)

C₂ -3C₃= 12-------------------(2)

3C₃= -9

C₃= -3

From equ 2  C₂ +9=12

C₂=3

From equ 1  3C₁ +3 - 9= -3

 C₁=1,  C₂=3, C₃=-3

[tex]-3+12x-9x^{2}[/tex]= (3)(1) + (1+x)(3) + (3-3x+[tex]3x^{2}[/tex])(-3)

zubam2002

Answer:

a) 3√11

b) C: (3x - 1)(x - 1) = (3)C1 + (1 + x)C2 + (3 - 3x + 3x²)C3

c) C = {3i, 2j, 3k} or C = {3i, 4/3j, 5/3k}

Step-by-step explanation:

a) C = {3, 1 + x, 3 - 3x + 3x²)

For x ≤ 2, C = {3, 3, 9)

P2, the vector space, C = {3i, 3j, 9k)

∴ P2 = √3² + 3³ + 9² = √99 = 3√11

b) - 3 + 12x - 9x² = 0

(3x - 1)(x - 1) = 0

x = 1 or 1/3

C: (3x - 1)(x - 1) = (3)C1 + (1 + x)C2 + (3 - 3x + 3x²)C3

c) -3 + 12x - 9x² ⇒ C = {3, 1 + x, 3 - 3x + 3x²)

From -3 + 12x - 9x²= 0

(3x - 1)(x - 1) = 0                                

x = 1/3 or x = 1

For x= 1

Cordinate vector, C = {3i, 2j, 3k}

For x= 1/3

Cordinate vector, C = {3i, 4/3j, 5/3k}

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