1 1 1 w 3: Let A = Show that A is invertible if and x2 y? z2 w² y 23 w3 only if x, y, z and w are distinct. 1.

Q: 0 0 0 1 0 0 0 1 0 ) Let A =

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Q: B) Suppose A is an invertible matrix inM, (R). And x, y E R". Prove it: det(A+ ry") = (1+y"…

A: Given: A is an invertible matrix in Mn(ℝ) x,y∈ℝn To prove: det(A+xyT)=(I+yTA-1x)det(A)

Q: Now assume that f and g are both C2. Express the matrix of second derivatives V²F(x) in terms of f…

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Q: [1 2 Define a linear operator T: R³ →R³ by T(u) = Au, where A = |2 li 3 2 31 3. 3 21 a. Find T(e1),…

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Q: 1. Let f : R° → R³ be f(x, y, z) = (x² + yz, y² + xz, z² + xy). (a) Compute Df(x, y, z). Is f…

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Q: Suppose u1, ūz and ūz are linearly independent. Let īj = ủ1+ ủ2, 02 = ủ1 + ủ3 and ī3 = ū2 + 2ūz.…

A: Given: Let u1→,u2→,u3→ are linearly independant. Let v1,→=u1→+u2→,v2,→=u1→+u3→,v3→=u2→+2u3→

Q: 1-x 0 <x< 1 (2) Find all the points of discontinuity of the function S(x) =}x+ 2 (x² – 4 1 <x < 3 x…

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Q: (b) Let X - N4 (u, E) with 2] 4 2 -3 4 4 2 4 1 and Σ- -1 -3 4 - 2 3 4 1 -2 8. Let Y and Z be…

A: i) X ~ N4(μ,∑ ) E(Y|Z) = ? Y|Z ~ N2 with mean μ1+∑12∑22-1(Z-μ2) Here,  μ1 = 24μ2= -13 ∑12 = 4224∑22…

Q: 1) Find the norm of the following operators d) T:C[0,1] → C[0,1] by (Tx)(t) = f¿(t - s)*x(s)ds

A: Norm of operator gives the idea of size of any linear operator. Norm of can be found as .

Q: 13. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw||² %3D a. If X…

A: given a matrix X and a vector y of appropriate dimensions, define as  fw=y-X w2 a) if X is N×k ,…

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: Find the value of x for which A is not invertible: x – 1 | 1 A = x2 x +2

A: Consider the given matrix. A=x-11x2    x+2   If A is a invertible matrix then A=0

Q: [1 1 = 0 1 1 = Lo o 1 [1 -1 01 Compute T, in the three steps, using U and U-1 in U-1 = |0 -…

A: Given: U-1=1-1001-1001-1=111011001 T3=UTU…

Q: Pls help ASAP

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Q: 3. Show that the action of C4(2) operator gives Ca(2) (X1, Y1, Z1) = (-yı,X1, Z1)

A: Consider this four-fold rotation around the z-axis. The effect of this operation on x,y,zis given by…

Q: 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,…

A: The solution is given as

Q: (4) 10. Let H = {r+2x2, 1+ ka?, k + (2k + 3)r2} (a) For what value(s) of k is H linearly dependent?…

A: This belong to linear algebra.we check linear independence for the given vectors and find k so that…

Q: [#1] Exercise 7.46. Prove that if y = y' then y = ky' for some scalar k.

A: Given that γ≡γ' The y can be written as,  γ≡r mod n The y' can be written as, γ'≡ r mod n

Q: 3.4.10. (a) Given x E co, prove that there exist vectors xn E co such that ||x – xn|| → 0 as n → o.

A: As you have asked more than one different questions, we shall answer first question only..   For…

Q: 0 1 3. Let XT = [X₁, X2, X3, X4] with mean vector μ = 1 0 and E = 21 9 -2 2 0 -2 4 (X₁) X₂ (X(1))…

A: Please note that as per our guidelines I have answered the first three sub-parts. Please repost the…

Q: 1- Let X1, X2, X3 have the joint pdf given by 1 eita) ,-c∞ < x1,X2,X3 < ∞ -00 f(x1,X2 ,X3 ) = {…

A: Normal Distribution: A continuous random variable Z is said to follow Standard Normal distribution…

Q: 3. Let T: R³ R³ be the operator given by 00 8 1 0 T(v) = -12 6 υ. Determine whether T is…

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Q: Example 3.8. Forx = 1,2,3 and y = 1, 2, let the joint pmf of X and Y be defined by I+y fx,y(r, y) =…

A: XY X=1 X=2 X=3 Total Y=1 2/21 3/21 4/21 9/21 Y=2 3/21 4/21 5/21 12/21 Total 5/21 7/21 9/21 1

Q: (d) Let A E M3x3(R) and , ij, Z ɛ R* and they are linearly independent. Suppose that we have A = ,…

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Q: Example 3.9. For x = 1,2 and y = 1,2, let the joint PMF of X and Y be defined by ry fx.y(x, y) = 15…

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Q: 22 y? 2 u?. Show that A is invertible if and r y 23 w3 3: Let A = only if r, y, z and w are…

A: Given that the matrix A=1111xyzwx2y2z2w2x3y3z3w3 Show that A is invertible if and only if x,y,z and…

Q: (: 1 2 3 Let T : M2x2 (IR) → M2x3 (R) be given by T(A) = AB where B = 4 5 6 Let B = {E11, E12, E21 ,…

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Q: 2. Let V = Mn(R). Show that the symmetric bilinear form : V xV →R given by (X, Y) = tr(XY) is…

A: The complete solutions are given below

Q: Example 3.8. For r = 1, 2, 3 and y = 1,2, let the joint pmf of X and Y be defined by x+ y fx,y(x, y)…

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Q: What is the condition should be add to make y invertible. Select one: a. x= y and x#–y b. x+y and x…

A: (1) The matrix is invertible if its determinant is non-zero. Here, the given matrix is: xyyx So, det…

Q: 2. Consider an operator of the form (Lf) (x) = Ef(x;)g;(x) where a O for all a E [a,b] and for all…

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Q: 1) Verify whether the limit to infinity of n(n + 1)(n + 2)−1(n + 3)−1 is equal to one.

A: Limit of the sequence

Q: 2. Suppose f: E" E" is an isometry given by f(v) = Ae for some n x n matrix A, n2 2. a) Show that g:…

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Q: Q6. Let C[0,1] have the inner product < f,g 2 S,"f(x)g(x)dx and let fn= cos nx (n=0,1,2,3,....).…

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Q: 2 0 Let A = and define T: R²→R? by T(x)= Ax. Find the images under T of u = a and v = 0 2 - 6 T(u) =

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Q: 22. (a) Prove that [0, x] = [0, y] for all x, y E Z, x + 0, y # 0. (b) Prove that [ax, ay] = [x, y]…

A: (a) we have to prove that  0, x = 0, y for all x, y∈Z, x≠ 0, y≠0 proof as x, y∈Z, x≠ 0, y≠0 Z is the…

Q: 2. (1) Consider the following function of two choice variables Z=f (xy) Write down the first order…

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Q: Q3: Using Fredholm determinates, find the resolvent kernel of the kernel : k(x, t) = sin(x) –…

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Q: 7. Find the Jacobian matrix of the following. a) x= (u+v),y = (u-v).= = 2uvw b) x=r cos 0, y =r sin…

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Q: 2. Let X ~ U[0, 1] and Yn = X. Prove that Y, 4x.

A: Given that, X~U0,1 Yn=n-1nX

Q: -52 +1 -3 If zo =-1, G(z) = and f (y) = -5 y1 4 - 2 y2 Y3 + v -4 y1 - 2 32, find the Jacobian of G…

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Q: 1 2 1 0 Find a mat rixP such that P- 0 3 0 3 1 | OA. 0 1 1 1 P O B. %3D 0 1 1 0 %3D C. 1 1 1 0 %3D…

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Q: Let z = 3 + 2i and w = 6 – 1i . Find out the following in the x + iy form. 1 i) WZ ii) . z-1 – w-1…

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Q: 1 20 В -2 1 x = с 3 –10 X, 1 3 a) Calculate the pseudoinverse B* . X1 for Bx = c, call it x. X2 b)…

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Q: The value of a' for which f : R → R, ax +7 flx) is invertible lies in %3D 2 x* +4 (а) (0, 2) (b) (1,…

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Q: H n A = Let - 1 2 -3 3 -1 0 T Compute the determi: on to triangular form. Determine whether th Ax =…

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