Define T by T: Pn → R by T(p) = =S² p(x) dx, where P is the set of all polynomials whose degree less than or equal to n. Find the n kernel of T.

Q: Compute the convolution L ★ II of the functions L and II, which are defined below: 1 if |æ| <a/2…

A: For the function, Lx and πx, Convolution of Lx and πx can be calculated by L∗πx=∫0x Lt πx-t…

Q: Let R be an integral domain and let a, b, c € R. Suppose that ab = ac with a 40. Prove that b = c._

A: Since, the image contains two different problems, according to the guidelines, I am solving the…

Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

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Q: Consider the map T: P2 → P1 given by T(ao + aj x + az x²) = -ao – a2 – 2aj x. (a) Calculate T(2x –…

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Q: 3. The linear operator T: R³ R³ is defined by T(x1, x2, x3) = (W1₁, W2, W3), where w₁ = 2x₁ + 4x2 +…

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Q: Let A = ū= [¹²3] a [¹3] and v u= LO [2] and define T: R² → R² by T(x) = Ax. Find the images under…

A: The given matrix  and the linear transformation .The aim is to find the images of  and .

Q: Let B = {1, t − 8, (t − 8)²} : - and C = {1, t − 5, (t − 5)²} - be two bases for P2. The function…

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Q: Let B = {1, t − 3, (t − 3)² } and C = - - {1, t − 2, (t − 2)²} - be two bases for P2. The function…

A: a=1b=-2c=5Explanation:

Q: Prove that xJ(x) = vJ , (x)– x (x) X V. v+1

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Q: 6A Illustration is given f: R³ →R³: (x, y, z) → f(x, y, z) = (x-2y, y-2z, -x+4z) Prove that f is…

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Q: Find real numbers x, y and z such that (x + 4)i + (y – 5)j + (z – 1)k = 0

A: To Determine :-  The real numbers x , y and z  such that   x + 4  i +  y -5  j +  z - 1  k =0

Q: The domain of the function g(t) = vt – 8 is O (8, 00) O [8, 0) O (-∞, –8) O R O R-{8}

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Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

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Q: Construct the Lagrange interpolation Polynomial for the function Y= Sin Tx, the ponts =0, 2, =…

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Q: Let T: R² R2 be a linear operator defined by Find a formula for T I Y T[H]-[2] + [2] = [3]. 5

A: To find the formula for the linear transformation of [x y].

Q: Find the norm of f(x) = √√1-x² on the interval [- 1, 1] . √3 a|m O 2 P 3 2√3 3 2

A: A norm is a function that assigns a non-negative length or size to each vector in a vector space.…

Q: f(t) = {b 0<t<3 otherwise

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Q: Let LR R be a linear operator such that L X, )-| X1 then Ker(L) spanned by: Select one: a. (e2,e3)…

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Q: Consider the map T: P2 → P1 given by T(ao + aj x + az x?) = (3a0 – 3a,) – 2a7 x. %3D (a) Calculate…

A: Note: As per our guidelines we are supposed to answer only 3 subparts and the rest can be reposted.…

Q: Determine the Wronskian of the given set of functions and determine whether they are linearly…

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Q: each linear operator T on V, find the minimal polynomial of T. (d) V = Mnxn(R) and T(A)=At . Hint:…

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Q: All computations must be shown. 1. Let pi(t) = 1+ t², p2(t) = t +t² and p3(t) = 1 + 2t + t² be…

A: Consider the provided question, given,p1t=1+t2p2t=t+t2and p3t=1+2t+t2

Q: Let r(t) =(t, 1 − t,t2)Find the general formulas for T and N as functions of t

A: We have rt=t, 1-t, t2 Tangent vector is given as Tt=r't So, Differentiate the rt with respect t.…

Q: Consider the linear operator T:R3→R3, given by T(v)=A⋅v, where [image]: What is the minimal…

A: Consider λ is the eigenvalue of the given matrix A. Therefore, 1-λ2301-λ100-1-λ=0. Expand the…

Q: Define T by T: Pn → Pn-k by T(p) = dk p(x) dxk 9 where P is the set of all polynomials whose degree…

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Q: Find the Wronskian of the following functions. Determine if they are linearly independent or not.…

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Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

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Q: f : R → R where f(x) = 6x – 17 -

A: " Since you have asked multiple question. we will solve first question for you. If you want any…

Q: Let T: P₂- R² be defined by +T (PO) = [PⓇ]. (a) Find a polynomial p(t) in P₂ that spans the kernel…

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Q: d d Calculate[r₁(t) r₂(t)] and [r1(t) × r₂(t)] first by differentiating dt dt the product directly…

A: give r1(t)=9ti+2t2j+6t3k and r2(t)=t4k So  r'1(t) =9i+4tj+18t2k and r'2(t)=4t3k