b. Prove that the polynomials p₁(t)=1, p₂(t)=1-t, and p₂ (t)=1+t-t², form a basis for P₂.

Q: part (d) and (e)

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Q: Let B = {B]. A} mation x→→ Ax. =[%] be a basis for R2 and A = Find the B-matrix for the transfor-

A: Given that The transformation is

Q: 7. Prove that p, (t) = 1+t², p2(t) = t – 3t2, p3(t) = 1 +t – 3t, is a basis for P2.

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Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

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Q: 1. Let u = [1 0 1]",0₁ = [1 1+ T -i]¹, 1+i 1- - V₂ = [2i 1 0], and v3 V3 = [2 −1 1]². Let = V1, V2,…

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Q: 14. Let P₂ be the vector space of polynomials of degree ≤ 2. Let D = {1 + t, 1 − 5t, 2 + t²}. Check…

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Q: a' Find the value of a for which S forms a basis for R3.

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Q: Consider the bases for M2x2(R) given by a' : and a the stan- dard basis. For the transformation T :…

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Q: Let B = vector of 7 = E [V]B 0 1 00 {[%] [83] [28] [89]} 9 9 relative to B. -5 -2 || be an ordered…

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Q: 3. Let T : R² → R² be defined by T = X Y -x+y x-y

A: 3. Given that  be defined by:

Q: (e) Explain that Sį is a basis for R3 and that S2 is not. (f) Determine the coordinates of X1 U =…

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Q: 4. Let A = [1 1 1]. Find a basis of N(A) and C(A).

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Q: Let p1(t) = t and p2(t) = t + 1 be polynomials in P3. (a) Show that p1(t) and p2(t)} are linearly…

A: Given: p1t=tp2t=t+1 (a) We have to show that p1t and p2t is linearly independent. (b) We have to…

Q: Consider the basis S = {V₁, V₂} for R², where v₁ - (1, 1) and v₂ - (1, 0), and let T:R² R² be the…

A: Explanation of the answer is as follows

Q: 2

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Q: Consider the basis S - (V₁, V₂} for R², where v₁ - (1, 1) and v₂ - (1, 0), and let T:R² R² be the…

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Q: The set B = {1+t², t + t², 1 + 2t + t²} is a basis for P₂. Find the coordinate vector of p(t) = 1 +…

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Q: Let V₁ = A basis for H is V₂ 1. 4 6, V3 -8 5 - 17 ‚ and H= Span {V₁,V2,V3}. It can be verified that…

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Q: Let uj = U2 = V1 = V2 and va Let U = {u, uz) and let V = {v, Va Va). (a) Let eju +Cu2 Solve for e…

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Q: Prove that y R³ as a linear combination of (5) Is a basis of R³ {0·0·0)}

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Q: Find a basis for the vector space {fE P3lf(0) = 0}.

A: As per the question we are given a vector space V (polynomial space) containing 3rd degree…

Q: 4). Find all values of a for which the given vectors form basis for R' -(:) --() ---() a? u = and w…

A: A set B is said to be basis of any vector space if (1) it is linearly independent set. (2)B spans…

Q: Let U = p E 4 (R): 1 -1 p = 0}. Find a basis of U.

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Q: Let B = {1, x, x², x³} be an ordered basis for P3. Find the coordinate vector of ƒ(x) = −4x³ + 1x² +…

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Q: 2. Let ū = (-1, 2, 1), ở = (1, – 1,–1) and w = (2, 3, 0). Find the coordinates of (2,4,4) with…

A: Using gauss Jordan elimination method to find the value.

Q: The set B = = {1 − t², t+t², 1 + 2t-t²} is a basis for P₂. Find the coordinate vector of p(t) = 5 +…

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Q: The set B = {1+t², -t+t², 1+2t+t²} is a basis for P₂. Find the coordinate vector of p(t) = 5-6t+13t²…

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Q: The set B= {1-t,t-t,1- 2t-t} is a basis for P2. Find the coordinate vector of p(t) = 1+ 5t – 4t…

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Q: Let T : R² → R² by T(x) = A˜, where 1. Find ker T 2. Find a basis for im T. A-[¹₁2] Α = -3 -6

A: 1) Ker T Solution:   1 2     -3 -6   Now, reduce the matrix to reduced row echelon…

Q: Find a basis of solutions by the frobenius method. Xy'' + 2y' + xy = 0

A: To find a basis of solutions for the differential equation xy″+2y′+xy=0 using the Frobenius method,…

Q: Consider the basis S = {V₁, V₂} for R2, where v₁ = (1, 1) and v₂ = (1, 0), and let T:R² → R² be the…

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Q: (d) What is the nullity of [T]? (e) Give a basis for the kernel of T.

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Q: Show that B = {v1, v2, v3} is an orthogonal basis for R3: v1 = (3,−3,0), v2 = (2,2,−1), v3 =…

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Q: 3. Suppose that {v1, v2, V3} is a basis for V and p:(v;) = dij. Let %3D W1 = 2v1 + v2, W2 = v1 -…

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