12. Let B = {1+2x, x – x², x + x²}. (a) Show that B is a basis for P2. (b) Let p(x) =1+3x + x². Compute the coordinate [p(x)]g of p(x).

Q: Which of the following sets of polynomials form a basis for P₂? (i) P₁ = 1 + 4x + 6x², P2 3+x = (ii)…

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Q: Let f(x) = −4, g(x) = -5x+7 and h(x) = - -2x29x+9. Consider the inner product (p,q) = p(-1)g(-1) +…

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Q: Consider the following ordered basis for P₂ (R): B {x²+2x+1, x + 1,1} [] 3 2 (a) What quadratic…

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Q: the basis of P2 cons tive to B.

A: Introduction: For some n, consider V to be a subspace of Rn. A basis for V is a collection of…

Q: Let P3 = {ao+a₁x + a₂x² + 3x³|ao, a₁, 02, 03 € R} be the vector space of polynomials of degrees at…

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Q: Let S = {1+ 5x, 1 – 2x-}. Which of the following polynomials could be added to the set S to form a…

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Q: Show that λis a root of char(x) if and only if there exosts a non-zero vector v in V with L(v)=λv

A: Assume that λ is a root of charL(x) then | λIdv–L | = 0. It is known that, the system of Bx = 0 has…

Q: Q3. Determine the kernel and range of each of the following linear operators on R³: (a) L(x) = (X3,…

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Q: 3) Let C be a basis that contains vectors (1 + (-1) x + 1x² + 1x³), (1 + 2x + 2x² + (−1) x³), (1 +…

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Q: 2 3 [+0 4 2 b = -2 Select true or false for each statement. Let A = and c = [12] 1. The vector c is…

A: The objective of the question is to determine whether the given vectors b and c are in the kernel…

Q: Consider the polynomials p₁ (t) = 2 + 5t, p₂ (t) = 2 − 5t, and p3 (t) = 4 (for all t). By…

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Q: Show that (2 – 2x², 3x – 1, −1) is a basis for P. 3z

A: A set of polynomials is called the basis of  if it has at least 3 linearly independent at most 3…

Q: Find a basis for the solution space  x1+3x2+4x3=0 3x1+8x2+7x3=0

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Q: Let A = - 7 -6, b 3 2 = [4].. and c = Select true or false for each statement. [1] 1. The vector c…

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Q: hich of the following sets of polynomials form a basis for P2? 1, p2 1- 5х, рз || (i) p1 х, р4 (i)…

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Q: Define the linear transformation T : P2 → P2 according to T (p(x)) = p(2x − 5). (a) Compute the…

A: Given T(p(x)) = p(2x − 5)

Q: asis: The set B = {2-3x², 8+x-12x², 33x² - (21+ 2x)} is a basis for P₂. Find the coordinates of p(x)…

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Q: The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z +…

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Q: Let B = {(1, 1, 0), (0, 1, 1), (1, 0, 1)} and B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} be bases for…

A: a)b)c)d)

Q: `

A: Step 1: Step 2: Step 3: Step 4:

Q: 4. In the vector space P3(R), let S = {p1(x), p2(x), p3(x)}, where P1 (x) = 63x° + 19x² – 37x – 41,…

A: (a) we see, p2(x) = -3p1(x), so S is not linearly independent. [Ans] (b) As S is not linearly…

Q: (a) Prove that for every positive integer n, one can find n + 1 linearly independent vectors in…

A: (a) To prove that for every positive integer , one can find linearly independent vectors in…

Q: Consider the basis B= {1, x+x2, 2x2-1} for the vector space P,. The coordinate vector of the…

A: Given that the basis B={1,x+x2,2x2-1} for the vector space P2. Here we have to find the coordinate…

Q: Let X₁ = [1 3 −1 1]²‚×₂ = [3 −1 3 1]′, and x3 = [-1 2 −3 1]. Is b = T an element of span{X₁, X₂,…

A: No, we can write it.

Q: Find a basis for the vector space of all quadratic polynomials p(x) = ax²+bx+c, that satisfy p(−1) =…

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