Let T: M22 → P₂ be defined by T T([1]) ¹([8]) T([1₂8]) T([i]) −3x² + (−1)x+ (1). :52+(−1)2+(1) MDB (T) = b₁ Choose the ordered bases B of M₂2 and D of P₂ such that [1 0 0 0] 0 1 0 0 0000 b₂ = b3 = B = {b₁,b₂, b3, b4 } where [1] [J] -1 0 b4 10 O = and D || = -1 (a 1x² + 1x - 1. = 1x² -1x + 1. =

Q: Simplify generator of I = (x – 2y, Y – 3z, z – 4x) as ideal of R[r, y, z], respectively of J = (x –…

A: Prove that all the variables belong to I. Then conclude the value of I.

Q: b) Find each integral 1) ((²-2)i-(t-2)²³ j+e¹k)dt 2) t (cosi+sin²j+3 2 2 i+3tk)dt

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Q: Find all homogeneous polynomials of degree 3 in R² 3 p = Ax² + Bx²³y + Cxy² + Dy ³ that are…

A: Let, the homogeneous polynomial beThen, the function p is called harmonic if

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A:

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…

A:

Q: Let f(t) = (t – 2)*(t+1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: Find the standard matrix of the linear mapping L ab sin (a) f Əx ∞ α 52 5 2 x2 23. [2271²-1472²7223]

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Q: =1 using Cauchy's Residue theorem.

A: Cauchy's Residue theorem: Let fz be analytic within and on a positively oriented simple closed…

Q: 2. Let B₁ be the basis 2²e², re², e² of V = span(2²e², zeª, z), and B₂ be the basis of that the map…

A:

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

A: 3. Given that  be defined by:

Q: Let A = [2] and C = [12] and X be the 2 x 2 matrix satisfying AX-2C = 31₂. Then a. X = NINMIY HIM b.…

A:

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: Let T: R³ R² be defined by T x1 4x1 - x2 = x2 x3 2x3 2 Let B = u₁ = 4 ,u2 = -5 , U3 = and C = { v₁ =…

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Q: The bases B = {u1,u2} and B'={u'1, u'2} for R², where and u'1= The transition mat rix PB-B' from B…

A:

Q: Consider the basis 5 = {vg. Vz} for R², where v, = (1, 1) and v, = (1, 0), and let T:R? – R2 be the…

A: Here we have considered the basis S = {v1,v2} for R2, where v1=(1,1) and v2=(1,0) and T:R2→ R2 be…

Q: 4. Determine the integral of f(z) = 1/[(z – a)(z – b)], where a and b are in general complex…

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Q: Find the representation of I with respect to (a) S; (b) T ; (c) S and T ; (d) T and S.

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 2. Consider a fourth-order polynomial approximation 0e (x) = a, + a2x + a3x2 + a4x3 + a5x* applied…

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Q: . Let B = [X³, X², X, 1] and C = [1, X, X²], which are ordered bases for P3(R) and P2 (R),…

A:

Q: Consider the basis S = (v, v2) for R², where VI = (-2, 1) and v2 (1.3). and let T: R2 R' %3D be the…

A: A detailed solution is given in step 2.

Q: Let k be the k-th coordinate functional corresponding to the natural basis of 1, that is k(x)=x(k)…

A: We have to show that φk is linear, we need to demonstrate that it satisfies the properties of…

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

A: Given that basis of R2, S=11, 10 And consider linear operator T:R2→R2, Tv1=1-3 and Tv2=-61

Q: [3 -2] Let A = 1 1 and define T: R² → R² by T(x) = Ax. Find |1 −1 -1 A. T ¹([₂]) = 2 [2].

A: Given: and define  by  we have to find

Q: -) Let A = [₁0], B = [32] 0 12 find Aª BT, BÃ AT and (AB)T. Is (AB)T = (A)T(B)T?

A:

Q: Use the transformation x = u² − v², y = 2uv to evaluate the integral 1 2√1-x √₁¹ √²√¹−x √x² + y²…

A: Given transformations are .The given integral is .The integral region in plane is drawn as follows.

Q: Let T: R³ → R³ be a linear operator and B = [v₁, V2, V3] a basis for R³. Suppose 0 T(v₁) =_=A T(v₂):…

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Q: Let the matrix of L: R³ R² with respect to the bases S = and -{-}} W₁ T= c. Find L W2 O be A= a.…

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Q: Exercise 4.3. Consider f = 3x“ + 2x +2 and g u and v such that fu + gv = gcd(f,g). = x° + 2x +1 in…

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Q: Find an integral basis for the field Q(-5+ V-28)

A: We have to find the integral basis for the field ℚ(-5+-28). So here our field is given in the form…

Q: Find invertible representation of Y, = 0.9Y;-1 + e; + 0.5e;-1

A: Given: time series AR(2) process  Yt=0.9Y(t-1) +et+0.5 e(t-1) Then find invertible of Yt?