3. Let B = Consider T : R2,2 → R²,² defined by T(A) = AB – BA. Let [o o] Bị and B2 :

Q: Let w = r = uv, s = u – 2v. Use appropriate forms of the p2+ s2" dw dw and du chain rule to find in…

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Q: 1. Let f(r1, 22, r3) = x1 sin(r23). Compute the following: %3D a) Dif(x1,x2, x3) b) D3f(x1, x2, x3)…

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Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = <x, 0>. C is any path from (a, b) to (c, d).

Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express zx andzy in terms of fu , fy , and the…

A: Given, Let z=f(u,Y) u=ax+byY=ax-by

Q: Prove Theorem (A + B )T = AT + B T

A: Prove Theorem (A + B )T = AT + B T

Q: Let T: P4(R) → P4(R) be defined by T(f(x)) = xf"(x). Find a Jordan canonical form of T.

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Q: Prove that if w=f(x, y, z), x=r-s, y=s-t, and z=t-r then dw dw dw + + = 0. dr ds ôt

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

A: Please see the answer innext step

Q: Vx E X, (x, x) ¢ R. X = {1,2,3}, h of the following relations R on X is irreflexive?

A: Here given set   X={1,2,3}   we know that relation is irreflexive when each x belong X (x,x) dose…

Q: Let r(t) = (t, 3 cos(t), 3sin(t))

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Q: Give the condition which ensure that Je-| < 1 where z E C.

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Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: by f(x) = (x2 + 1)35 for all x e Ris A. One-One but not onto B. onto but not one-one C. neither…

A: Given f(x) = (x2 + 1)35

Q: please solve this linear algebra problem.

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Q: Find the fowien senies fon | X), -T<X <A

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Q: Given u(x, t) = t-²e¯, 2 , compute:

A: We can find the required answer as below.

Q: [2 Lo 2. and defined T: R² → R³,by T(X) = AX. Let A Find the images under T of u = |E|, and v =

A: Given a linear transformation T:R2→R3 by T(X)=AX where A= 2002

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Let E=RandT:R R be a Linear tiansformation I(,)=(2,3)and Tls0)=(3,2) Check whether T exists or noti2…

A: Given T:ℝ2→ℝ2 is a linear transformation such that T1,1=2,3 and T1,0=3,2. Now find the matrix of the…

Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express z x and zy in terms of fu , f y, and the…

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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: * Let A E R"Xn, ĩ E R" \ {0} and A = 0. Prove that if the origin is stable if and only if ī is…

A: Given A∈ℝn×n,x¯∈ℝn\0 and Ax¯=0 To prove that the origin is stable if and only if x¯ is stable.  If…

Q: Prove that i 224 s 2xP for xzO, y zo i and 1 t L-l with p>

A: Given that x,y≥0. And 1p+1q=1 with p&gt;1. Consider gx=xpp+1q-x. Then g'x=xp-1-1 and g'x&lt;0 when…

Q: () Let z = x + iy. Find Im and Im 2 in terms of x and y. Im Im z2 ||

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Q: 10. Let X , j= 1,.., n, be i.i.d. r.v.'s such that E(X,)= µ, o'(X)=o', both finite. j' Show that E(X…

A: The normal distribution: A continuous random variable X is said to follow normal distribution if the…

Q: Prove that H(x, y) = (x, –y) provides a conjugacy between x' = (4 })x and Y'= (; 7). X and Y' = 1 -1…

A: Given: X' =11-11XY' =1-111Y To prove: H(x,y) = (x , -y) provides conjugacy between both of the above…

Q: Let x, y E Rn be given. (a) If ||x|| = ||y|| = 1 and x y = 1, show that x = y. (b) If ||x|| ≤ 1 and…

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Q: Let r, y EN such that y # 0, then y+r#0.

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Q: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

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: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

Q: 1. Let TE L(V) and suppose v is in the range of T and w is in the null space of T". Show explicitly…

A: To solve we will use the definition of adjoint operator.

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Define T:R → R' by T(x, y, z) = (x-5y+4z, y- 6z). a) Is T one-to-one? Justify your answer. b) Is T…

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Q: Calculate A and B so that t B %3D 1+1 (1-z²)

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Q: Let T be the linear thansformation from R³ to R4 defined by 1 -8 1 X1 1 3 2 X2 X2 -2 2 4 X3 X3 3 0…

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Q: T: M22 ---&gt;M22 defined by T(A) = AB,

A: Given, a matrix B, and defined a map T(A) = AB, where T: M2×2  ---&gt; M2×2. We have to find the…

Q: . Suppose that T e L(V) and U and W are invariant under T. Prove that U + W is also invariant under…

A: If U is subspace of V and T(U) is subeset of U then U is invariant under T

Q: Prove that ||ư + B|| = ||T|| + ||B|| if and only if i = kủ. %3D

A:

Q: Find det (T) for T:R- R, where T(x,,×2,*3) = (x,- x2, X2 - X3, X3-x).

A: We will find the matrix [T] from the image of the standard basis vectors of R³ under the…

Q: Let S-(xeZ vmod12-8) and T (xe2| 4x). Prove that SCT but TgS La s-(rez| 3reZz, x-12r+7) and T-(xeZ|…

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Q: 41. Let f(x, y, z) =x?y+ y²z. Use the Chain Rule to calculate af/as and af/at (in terms of s and f),…

A: Since f is function of x, y and z.  For partial derivative f with respect to s and t we using chain…

Q: Let S = {(x, y) :x² + (y – 1)² < 1} U {(3,0)}. %3D Is the origin an extreme point of conv S? Is the…

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Q: -x+y<y. 2 If x < y, prove that x <

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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: Let fons =loag, Cln). Find t'ot at

A: Given, f(x)=logx(ln(x))

Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

A: