2. Let T: P2(R) → R³ and 4 2 [T]; 1 1 1 where B = {1, x, x²} and y = Is T invertible? {(1,0,0), (0, 1, 0), (0, 0, 1)}, find T(1+x+x²).

Q: T(1,1)=(5,0) Let T:²²be defined as, T(1,0)=(2,6) Then, T(3,2)=(12,6) (a) T(0,1)=(-3,6) (b)…

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Q: Let r(t): = 1 t-1' +-1/2). Find the domain of r(t). et+1, t-1,

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Q: Given the modulus n, the encryption exponent e, and the plaintext M, use the RSA encryption to find…

A: Solution :                    We know by RSA :                                     C = Me( mod n )…

Q: Q:2 If T1,T2: R² R? linear operators and are both definition by T;(x1, x2) = (x1, 0) and T2(x1,x2) =…

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Q: 6. Let p(t) = ² + 1 and g(t) = t - 4 be vectors in P2 with inner product (f(t), g(t)) = f(t)g(t) dt.…

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Q: 4. Let T: R2 → R² be defined by T = {B.[4]). and B₁ = -x y X- ([])=[ + ][²] = {[1].8]} BVB RUE BUE…

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Q: Let X=ℝ2 and define d,:ℝ2×ℝ2→ℝ d2((x1 ,y1),(x2,y)2)) = max{|x1 - x2|,|y1 - y2|}. Draw the…

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Q: Example 7. Show that T: E2 → E2, defined by is not linear T((x₁, x₂)) = (x₁ + x2, x1 - x₂ +1)

A: as we now for a linear map T(01 ) = 02  where 01 is zero of domain and 02 is zero of codomain

Q: Q.4. Let T is a transformation from R' into R’ defined by: T(x;,*;, X; ) = (2.x, –x,, 2x; – 4x,, x;…

A: 1.)Linear Transformation:                Let U and V be two vector spaces over the same field F.…

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: If f(r, v)= +y-2y, find fa (2, 1), f (2, 1), fay(2, 1) and fy(2, 1).

A: Given a function f(x,y)= x3+x2y3-2y2 Now, fxx= ∂2f∂x2=∂∂x∂f∂x fyy= ∂2f∂y2=∂∂y∂f∂y…

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: d Calculateri(t) r2(t)] and [r₁(t) × r2(t)] first by differentiating the product directly and then…

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Q: Q2):- Find Y[Z] of signal Y[n]=(Cos wn). U[n]

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Q: 1. Show that the given vector functions are linearly independent on (-∞, ∞). X₁ (t) = = 0 X₂ (t) H…

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Q: (a) Find the constants A, B, C, and D that satisfy 1 1 1+ (w/o)² 1+((z-w)/7)² Aw B Cw D st: 1+…

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Q: fi(x) 1 A = {fi, f2, f3} c F(R) and f(x) = f:(x) = |x+ 1| PROVE: A is linearly independent. (A L.I.)

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Q: Determine the Jacobian matrix of the function f:R³ → R³ given by f(x,y,z) = (3x²y+ yx²,4x²yz², 6xz³…

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Q: Let y 1 = e3x and y 2 = xe3* . Which of the following statements is true about W(y 1, y 2) ? (a) W(y…

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Q: Let A(t) = f(t)i + f₂(t)) and B(t) = g(t)i+g₂(t)), te[0, 1] where fi, f2, 81, 82 are continuous fun-…

A: Solution

Q: Q.4) Find the Legendre transform G(v1, v2, w) of the function F(u1, u2, w) = 2u – 3u1u2 + u3 + 3wu,…

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Q: Exercise 2. Show that if u(x, t) = F(x+ct) +G(x- ct) satisfies u(x,0) = f(x), u(x,0) = g(x), for all…

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Q: 10. Let V = M2×2; {{: ] =v: {[" :]=v : acR} w,(R) = { = Jev : a, b, cɛ! W1(R) W2(R) = E V : a E R…

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Q: find the values of c that satisfy Rolle's Theorem

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Q: 2) a) Let T: C² C²: Ty = [x+2y-iz] 2y + iz Lix+z-2z Find [7] B [7] B and P where

A: Given information  T  :  C2 →C2T xyz = x+2y-iz2y+izix+z-2z simplify it, T xyz = x+2y-iz2y+izix+z-2z…

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=1, b=-2, c=5Explanation:Step 1: Step 2: Step 3: Step 4:

Q: 3) show that (N) = ENONP(N,V) = Nop and (N²) – (N)² = Nop(1– p) No N=0+ %3D hint : write N2 = N(N –…

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Q: The function p(t) = 5 + 7(t − 9) + 2(t − 9)², can also be written as p(t) = a + b(t− 3) + c(t − 3)²…

A: We have to find the value of a,b and c using the given data.

Q: Example 1.5. Let X = R? x R² and d1 : X R, such that, (1) di(x, y) = d;((r1, 12), (Y1, Y2)) =| ¤1 –…

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Q: -5x²y³ 3x4 + 7y6¹ if (x, y) = (0,0) if (x, y) = (0,0) (a) Show that g has an essential discontinuity…

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Q: 4) Find the Hermitian form [18] [2] X = H= XTAX where

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Q: fi(x) A = {fi, f2, f3} c F(R) and f(x) fa(x) 1 |x+ 1| PROVE: A is linearly independent. (A L.I.)

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Q: T is a linear operator on V, (R). B, = {(1, 1, 0), (1,0, 1), (0, 1, 1)} and B, = {(1, 0, 0), (0, 1,…

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Q: (b) By determining constants c,, C2, C3 and c4, which are not all zero and are such that cifi(x) +…

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Q: 7. Let V={(x,y) x,y € R}. Suppose addition and scalar multiplication are defined using the following…

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Q: 23)a)givenx(t) and y(t) as below (3) y(t) X(t 2 4 2 Oplot x 1-0.5t) i)plot y(2t+1) iii) find x((1–…

A: Given that:   To plot x1−0.5t and y2t+1 Also to find ∫04x1−0.5ty2t+1δt−1dt

Q: Let V = C[0,1], p(t) = 1-t and q(t) = 1+t. Compute (p, q) 19 19 (b) (c) (d) None of these 5

A: Given V = C[0,1]p(t) = 1-t2q(t) = 1+t2

Q: Let T: P2 → P₁ be defined by T(p(x)) = p'(x). Let B = {1 + x + x², 1 + x, 1} and C = {1+ 3x, 2+5x}.…

A: For the given transformationto find the matrix representation of T with respect to basis B and C.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.