Let P² be the vector space of all polynomials with real coefficients of degree at most 2, with the usual operations. a) Show that the set of vectors S = {x² + 1, 3x − 14, -5x² - 9x +37} is linearly dependent. b) Find a basis for P² which includes the first two vectors of S (that is, v₁ = x² + 1 and v₂ = 3x – 14).

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: The set B = {2 - 2x², 6 - 2x − 6x², 13 - 6x - 12x²} is a basis for P₂. coordinates of p(x) = 18x −…

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Q: The set B = {3+4x², 12 − 3x + 16x², 9x - 25 - 36x²} is a basis for P₂. Find the coordinates of p(x)…

A: Here we have to find the coordinates of p(x) relative to the basis B.

Q: The set - B = {−2+2x², −6+ 4x + 6x², − 6 + 8x + 8x²} is a basis for P3. Find the coordinates of p(x)…

A: Given basis is .Also given that we have to write down it as linear combination of basis elements,Now…

Q: Let B be the basis of P3 consisting of the Hermite polynomials 1, 2t, -2+4t², and -12t+8t³; and let…

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Q: The set B = {4x²3, 12x² (9+x), 38+ 3x - 48x²} is a basis for P₂. Find the coordinates of p(x) = 44+…

A: Given; B=(4x2-3, 12x2-(9+x), 38+3x-48x2) p(x)=44+3x-56x2

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: The set B=(3+2x², 12+2x+8x²,- (29+6x+20x2)) is a basis for P2. Find the coordinates of p(x) = 8 + 4x…

A: Solution :-

Q: What is the basis and dimension of M3(R)?

A: Basis: For a vector space V, the set of linearly independent vectors that generate the whole vector…

Q: Using the scalar form of the table 1 3 2 3 1 1- - a 1 6 = 6+ a 5 4-a 2 -2 5 12 19 to determine the…

A: Step 1: Write the matrix A in scalar form. A = [[1, 3, 5, 6 + a], [2, 3, 4 - a, 2], [1, 1 - a, - 2,…

Q: The set B = {4+x², 20 - 3x+5x2, 45-6x + 11x2} is a basis for P2. Find the coordinates of p(x) = 6x -…

A: 30-1Explanation:

Q: Let B be the basis of P, consisting of the three Laguerre polynomials 1, 1-t, and 2- 4t +t, and let…

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Q: Find a basis for the ideal √(x³ - 214 + 2x² - x, x5-14-21³ + 2x² + x − 1). ស

A: We have to find a basis for the ideal x5-2x4+2x2-x,x5-x4-2x3+2x2+x-1

Q: EX 7 Using the scalar form of the table A /1 3 5 2 3 1 1-α 1 6 4-α -2 12 6 + a 2 -5 19 to determine…

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Q: *) Let P₂ be the vector space of polynomials of degree 2 or less. Consider the following two ordered…

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Q: The set B = {4x? - 1, z -3+ 12z2, 12 – 3x 40x} is a basis for P2. Find the coordinates of p(x) = 52…

A: write each polynomial as a vector  4x2-1=x2x140-1 12x2+x-3=x2x1121-3 -40x2-3x+12=x2x1-40-312 write…

Q: The set B = {3 + 2x², 6 − 2x + 4x², 4x − 13 – 10x²} is a basis for P₂. Find the coordinates of p(x)…

A: The objective of the question is to find the coordinates of the polynomial p(x) = 18x - 54 - 40x^2…

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

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Q: Let T: P2 → P2 by T(ax? + bx + c) = (12a + 5b – 5c)x² + (-20a – 8b + 10c)x + (10a + 5b – 3c); a the…

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Q: Consider the polynomials p₁ (t)=3-t³, p₂ (t)=2-t², p₂ (t)=t-5t², and p₁ (t)=7-t+3t² −1³. 3 Is {P₁,…

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Q: Consider the basis B3 = (x² - 1, x, 1) of P2 (R). Find [2x² + 3x − 5] B (2,3,-5) (2,3,-3) (3,2,-5)…

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Q: Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to…

A: Solution is in explanatio area.Explanation:

Q: Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to…

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Q: The set B = {− (1+3x²), − (2+2x+6x²), − (6+4x+21x²)} 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: The set B = {2 + 3x², 10 + 4x + 15x², 26 + 8x +36x²} is a basis for P3. Find the coordinates of p(x)…

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Q: The set B = {3 − 3x², 15+ 4x − 15x², 48x² - (47+ 12x)} - - is a basis for P2. p(x) = 12x² [p(x)] B…

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Q: Let A = 2 3-6 1 −1 3 3 - 2 Find bases for the row space and column space of A, and find the rank of…

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Q: Let B be the basis of P, consisting of the three Laguerre polynomials 1, 1-t, and 2-4t+t, and let…

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Q: Which of the following sets form a basis for the vector space P₂(R) = {ax² + bx + c : a, b, c = R}…

A: According to guidelines first three subparts can be solved.

Q: Consider the quadratic polynomial q(x) = 2x + 8x1x2 – 8x – 3x1 – 3x2 + 3. - - Find the symmetric…

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Q: The set B = {4- 2x², 12 + 3x − 6x², 18x² - (35 + 6x)} is a basis for P₂. Find the coordinates of…

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Q: B = {3 – 3x, 9 – 2x – 9x², 13 – 4x – 12x?} is a basis for P2. The coordinate vector of p(x) = 4 + 2x…

A: We have to find the coordinate vector of P(x)=4+2x-6x2. w.r.t the basis…

Q: The set B = {2 + 2x², 4 − 3x + 4x², 19 − 9x + 18x²} is a basis for P₂. Find the coordinates of p(x)…

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Q: Show that B = {1,1 – x,2 – 4x + x²,6 – 18x + 9x² – x³} is a basis for P3(R). Hint: Recall that…

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Q: Consider the following "standard" basis of R³ over R: --000 = Let q be the real quadratic form (from…

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