Exercise 2. Prove that if A = B and C z D, then A × C = B x D.

Q: Show that in R2, (a) i·i = j·j = 1; (b) i·j = 0.

A: a)- To show that i·i=j·j=1 As, the dot product is represented by: a·b=abcosθ Using, it we have:…

Q: (b) If z r+ry- y7, 1= u+ and y = 2uv, find: in terms of du %3D u and v.

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Q: following (a) E(X) and E(Y) (b) Cov(X, Y) (c) Are X and Y independent? Justify your answer.

A: Solution

Q: Show that ex and e−x are linearly independent in C(−∞,∞).

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Q: (b) If V'u =0 for x2 0 and if u f(y) on x=0, Show that ,y): +(y-5)

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Q: 1. Show that S, has the presentation (x, y|x² = y = (xy)² = 1). %3D

A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

Q: Given, f(x, y, z) = xln(x?y)+ x*y – e a2. Compute fa, f, fry, fayz-

A: If f(x,y,z) is a multivariable function, then the variables x,y, and z are called the input of the…

Q: Say that E = Ãek™, where the components of A and k are real and constant, i.e. they aren't imaginary…

A: A vector is a quantity that has both magnitude and direction. In general, vectors are represented as…

Q: et T E L(V,W), and U e L(W, X). Prove that UT : V → X is also linear

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Q: "en J(2, 9) 5y), find =(z, y) = (z, y) = -(z, y) = %3D

A: We wil find the second order partial derivative.

Q: Assume X, Y CU. Prove that (U \ X) U (U \ Y) = U \ (X n Y)

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Q: Let -2xy – 3yz – cz, x = st, y= e", z=t? Compute ди (4,-5) = ds dw (4,-5) =

A: The given function.

Q: verify that (A + B)T = AT + BT

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Q: Solve min udr, subject to u(0) = 0, u(1) = 1, ur'dr = c %3D =C.

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Q: Find the uncertainty in U, assuming that X = 10.0 ± 0.5, Y = 5.0 ± 0.1, and a) U = XY² b) U = X² +…

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Q: Find solutions of the linear PDE u, + yu, =0 , given that u(0, y)= y'. ди(х, у) ди(х, у) Notation is…

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Q: A- If y = e3x and y = e-3x , prove that: - d²y dx2 – 9y = 0

A: The solution for the above question is as shown below.

Q: Prove or disprove: Vx € Q, Vy e Q, 3z E Q, (x < y) → (x < z< y).

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Q: Let W = {a + bX + cX² + dX³; a – b = 0, c = 4b,}. Then the dimension of W is %3D

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Q: Let X and Y be iid N(1,3). Let U = 4X + Y and V V_dependent or independent? Justify your answer. 2X…

A: Given X  and Y be iid with N(1,3). Mean, μ=1 Standard deviation, σ=3  U = 4X + Y and V = 2X — 3Y

Q: A If w= f (x, y), where x = rcos 0 and y = rsin 0 , show that 2 1 + 2 +

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Q: If u -x2 + y? + z² be such thatr u,+ yu, + zu̟ = Àu, then A is equal to %3D + A.1 В.2 C.0 D. none of…

A: Given that, u=x2+y2+z2 Then xux+yuy+zuz=λu The aim is to find the λ value, ux=2xuy=2yuz=2z

Q: (b) If z =W, show that: x- y xy 0 =

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Q: Let x, y € R" be given. (a) If ||z||||y||=1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1 and ||y||…

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Q: How to solve this?

A: We are given that x + y + z = 5 where x, y, z ∈ R. Now, we need to prove that x2 + y2 + z2 ≥ 5/3.…

Q: Let a,b E R with a # 0, b # 0. Prove that (ab)¯1 = a°1b´1. %3D

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Q: Find and d?z and prove that 2 0²z IF 2 = f(xt ay) + F CX-ay) %3D

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Q: 1. If x, y, z E R, prove that (c) ||x| – |y|| < |x –yl. (d) \xy| < }(x² + y²)

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Q: Show that ∗ : R × R → R defined by        a ∗ b = a + 2b is not commutative.

A: Show that ∗ : R × R → R defined by        a ∗ b = a + 2b is not commutative.

Q: Q₂. if z=f(x+ct) + g(x-(t) then show that C²Z = Ztt XX

A: Given function: z=fx+ct+gx-ct We have to prove that, c2zxx=ztt. We know that, daxndx=anxn-1

Q: 3. Assume that z= 2(r.y) and ee = ryz Find and

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Q: Show that if X and Y are independent rv's, then E(XY) = E(X) · E(Y)

A: we have given that, x and y are independent random variables, then, we will prove, E(XY)= E(X).E(Y),

Q: Q- show that yo+(?) sy + (2) B²y_₁ = y + (1) DY...+ (1²¹) Ay -

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Q: If x = v'u and y = 2uv, show that JJ' - 1, where J is the jacobian of u,v with respect to x and y…

A: We have find the jacobian

Q: Let w = 4x? + y? + z? x = psinpcose y= psinpstno z = pcoso Find dw dw dw de ee de

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Q: Prove that Àx (Bx)= A.¿)B CA.B) wheve. A.a,t.azj+a3k. .....

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Q: Suppose T(x,y) = (2x, 2y) and U(x,y) = (x, -y). Is T∘U=U∘T?  Justify your answer.

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Q: Determine C, and C, so that y(x) = C, e²* + C,e* + 2sinx will satisfy the conditions y(0) = and…

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Q: Given that z = f (u,v), u=x + y and v = 2x - 2y. Show that 4. дхду ди?

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Q: 8 Exercise.3 Evaluate y* Va? – y² dy [+hint: assume y² = a?x], [M Check answer: %3D 32

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Q: Question dz If z(x, y) = x²y, x = ", and y = u'v², find

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Q: Prove that B (xX+1; y ) + B (x,y+1)=B (X,Y) let П (х+1) = ХГ (Х)

A: We have the formula for beta function interms of gamma function is that B(m,n)=Γ(m)Γ(n)Γ(m+n)As we…

Q: Construct a proof of the following fact: (Z V T) → P, Z, (P V R) → =(Q V T) t -(QV T).

A: Given:                                                           Z∨T↔P,Z,P∨R→¬Q∨T⊢¬Q∨T To do:…

Q: If f(x, y, z) = In(2 + x² – 2y² – 3z), 7 =, then Duf(2, –1, 1) = - A) * V3 B) 11 C) V3 D) E)

A: The given function is:    f(x,y,z)=ln2+x2-2y2-3z

Q: Show that in R2 ,(a) i · i = j · j = 1;           (b) i · j = 0.

A: We have to show that In R2 (a)       i · i = j · j = 1   (b)      i · j = 0

Q: Q/ z = x + f(a) and a = ry. Show that: 尝-端=

A: We use the definition of partial derivatives to prove the given result.

Q: If F(x,y, z) = (5x – y + az)î + (bx + 2y + 5z)ĵ + (2x + cy – z)k is irrotational then find the value…

A: Explanation of the answer is as follows

Q: O eiz dz < z2 – i e2 , 3 - z(t) = 2e", t e [0,27]. true False

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Q: If p(x, y, z) = x²z + ey/x, evaluate grad op at the point (1, 0, -1).

A: The function is φ(x, y, z)=x2z+ey/x