Prove that if A−1exists, then there can be no nonzero y for which Ay = 0.

Q: 5. Prove that So*(y – 0)dydr =

A:

Q: Бxї Zet (x, г) бе space, then Co-countable тебя xis not Тз-брасе

A:

Q: Let x, y ∈ R. Suppose x^2 +y^2 = 0. Prove that x = 0 and y = 0.

A:

Q: y If z = show that = 0. x + y?*

A:

Q: Find the Wronskian of a given set {yı, y2} of solutions of 7. (1 x2)y"-2xy' (a 1)y0, _

A: Consider the given equation.

Q: 3. Find all the fixed points and classify cach of them: f(r,u) ryev,

A: Solution:- Fx,y=xye-x2-y2 Put Fx=0 and Fy=0 Fx=e-x2-y2y-2x2=0and Fy=y-2y2 e-x2-y2=0 So,  0,0,12,12,…

Q: Q4 If f(x, y) = In(x² + y² ) %3D Prove that dy2

A:

Q: Prove that if Y:nz+P(x- mz) So thdz

A:

Q: Prove that X∩(Y−Z)=(X∩Y)−(X∩Z)

A: We prove that X∩(Y−Z)=(X∩Y)−(X∩Z)

Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: if JERLA.b] and it hex)- J(x). except for finite mumber of Paint in Cab] then that he Rlab] prove.…

A: if g ∈R[a,b] and if h(x) = g(x) except for a finite number of points in [a,b] then prove that h…

Q: y Verify Euler's theorem for u = x' In

A:

Q: State where in the ty-plane the hypotheses of the Existence and Uniqueness Theorem are satisfied for…

A:

Q: If y = xsinx then prove that y''+y=2cosx

A:

Q: (b)* Show that Y' is closed.

A: Actually we shall show, closure of the set = the set. Then this set will be closed.

Q: az Find if yz – In z = x + y ax 3.

A: We have              yz-lnZ=x+y

Q: Let f(x, y, z) = x²y°z – xyz², and compute: (a)

A: In this question we will use basic of vector  calculus i.e gradient of any function and curl of…

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||…

A:

Q: Prove Theorem (AT )T = A

A: Given: Theorem, ATT=A.

Q: Show that f: R → R defined as f(a) = 3a³- 4 is a one to one %3D iuicuon.

A:

Q: Prove that if x+y=%30 then y=-x

A: See the attachment

Q: Let x, y ∈ R. Suppose x2 + y2 = 0. Prove that x = 0 and y = 0.

A:

Q: Четверг show that the inu of a bounded E tix-oy ne 7 8. use T

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Q: b) = f ) then prove that x ux + y uy + z uz = 0 u

A:

Q: Determine if the following is a fundemental set of solutions to  y'' -6y' + 25 = 0 Given that…

A: If the Wronskian of two solutions is not equal to zero, then the given set of solution is…

Q: Proof by contrapositive that for xy EZ, if xy is even then either x or y is even.

A:

Q: of fxyl = 3X-y+6 Subjected QI Find to x+y=4 extrume v alues

A:

Q: Verify that the conclusion of Clairaut's theorem is valid, that is, that uxy = uyx u = x4y3 - y3…

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Q: ху • prove that x +y Example If z = az dz + y.- Əx х. ду

A:

Q: Prove that H(x, y) = (x, –y) provides a conjugacy between 1 X' = 1 X and Y' Y. 1 1 1 1

A: We are Given that :  H(x,y)=(x,-y) X'=11-11X , Y'=1-111Y The Matrix H is corresponding to the…

Q: If (x* – y*,x – z°) = 0,z = z(x, y), prove: az y3 az +x³z4 ax - %3D ду 3

A:

Q: If Z = f(x + g(y)) , prove: Zx * Zyx = Zy * Zxx

A:

Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

A:

Q: • Verify that 3r is a one-parameter family of solutions of the DE ry + y = 3r. • Deduce a solution…

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Q: find the root of the fnmetion ーX FCx)= ē^ _ x which lies betwen (o %3D

A:

Q: Q3/ prove fry = fyx if f(x,y) = x + xy – 5x³ + In(x2 + 1)

A:

Q: Prove that if X₁X and Ync, then (Xn, Yn) (X, c).

A: For a sequence of random variables X1, X2, X3,... to converge to a random variable X, we must have…

Q: 3. Show that à = (6xy + z³)î + (3x² – z)ĵ + (3xz2 – y)k is irrotational. Find p such that A Vọ.

A:

Q: /show that fo-1x is-Contindals everywhere.

A:

Q: Prove that (X+Y) (X+Z) =X+YZ

A:

Q: For Z= x²csc y,prove that Zxy is equal to Zyx

A: Given that,    z=x2csc y

Q: if y ". y (In z) (a) Find

A:

Q: Find the absolite marima and minima points between [o,u] gf y : * + 2dx' sunchion is porsbl.

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Q: Given z = , show that ху xy(Zxx + Zyy) у3 — х3 x²y2 2.

A:

Q: Suppose that T :C C is R-linear. Show that T is C-linear if and if T(iz) = iTz for all z.

A: First we describe the definition of C-linear function:

Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

A:

Q: Prove that if X ⊆ Y then X ∩ Z ⊆ Y ∩ Z.

A: To prove this we will take an element from X∩Z and show that 'a' also belongs to Y∩Z.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.