Show that if A is invertible and DA = CA, then D = C.

Q: If A is invertable then A is diagonalizable. Select one: O True O False

A: If a matrix A of order nxn is invertiable then it has n disctinct eigen values.  And corresponding…

Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

A:

Q: Prove that Cov (x, y) = E (xy) – E (x) E (y).

A: Cov(x,y) is covariance of x and y. E(x) is expected value of x E(y) is expected value of y.

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

A:

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

A:

Q: Show that the operators T and T₂ from 2 R² into R² de fined by T₁: (F₁) √₂ ) → (9₂9₁) and T2=…

A:

Q: det(A) Show that det(AB-') = %3D det(B)

A: Consider, Let A and B be the two n×n matrix, Now Consider, detAB-1 We know that the theorem, Let A…

Q: Check whether the function f(2) = e5² satisfies the Cauchy-Reimann equations or not.

A: 2) The given function is f(z)=e5z. Check whether the function f(z)=e5z satisfies the Cauchy-Riemann…

Q: please solve this linear algebra problem.

A:

Q: Let A ∈ M (3,R) be invertible and C, D ∈ M (3,2,R) . Solving B from the equality A(B+Ct)t = D, what…

A:

Q: show that the scalar t that minimizes ||y– tx||

A: In this question, we need to find the t that minimizes ||y-tx||.

Q: a) Let f: Z Z be such that f(x) 2*x. Is f invertible, and if so, what is its inverse?

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: find the linearization at x = a. f (x) = x4, a = 1

A: Given function: Using the formula of linearization of a function f(x) at x =a :

Q: Show that: (a) Cov(X,Y) = E[XY] – E[X]E[Y]. (b) If X and Y are independent, then fY|X(y|x) = fY(y)

A: Solution:a.Show that COV(X,Y)=E(XY)–E(X)E(Y)

Q: #2. Find invertible matrxes B and C SO C M.B =

A:

Q: Looking Ahead Use the Wronskian to show that {e' , te' I is linearly independent on R.

A: Consider the equation

Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

Q: Let f(2) := 25. Show that f is invertible in a neighbourhood of i and calculate (f-1Y(/(i)).

A: Let f(z) = z5, show that f is invertible in a neighbourhood of i and calculate(f-1)' f(i)

Q: find the linearization at x = a. y = (1 + x)^−1/2, a = 0

A:

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

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: Q3: a) Let ƒ : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its

A:

Q: Determine if the functions Y1=e^{4x}senx and Y2=e^{4x}cosx are linearly independent or linearly…

A:

Q: (b) Show that the Linear guen Operators. are not with T(f)= Lf6), f()+1].

A:

Q: The linear operator T: R2 --> R2  defined by T (x; y) = (4x + 5y; 2x + y) is it diagonalizable?

A:

Q: Show that following MA models are equivalent A: Yt = et – Oet-1 for et ~ WN(0, 1) 1 Yt = et for et ~…

A:

Q: 5. Conclude from this that if b x c = 0 then b is a scalar multiple of c or vice versa (b and e are…

A:

Q: Use Cramer's rule to solve for y without solving for x, z, and w. 5 x + y + z + W = -7 5x + 3y - z +…

A: Here we have to use cramer rule for solving the system of linear equations

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

A:

Q: Find the gradient of the following scalar fields: a) F=x^2+y^2 b) V=p^3z cos2phi

A:

Q: find the linearization at x = a. y = (1 + x2)^−1/2, a = 0

A: We find the derivative of the given function using power rule and chain rule

Q: Let A ∈ M nxn(F). Show that A is diagonalizable if and only if At isdiagonalizable

A:

Q: Let Te B(H) be an invertible operator. Show that a(T-) = 0(T)-.

A:

Q: Show that u = (a, b) and v = (e, d) in K? are linearly dependent if and only if ad – be = 0.

A: Solved below

Q: If A is invertable then A is diagonalizable. Select one: True O False

A:

Q: a) V = R", (x, y) = (Ax) · (Ay) where A e Mn,n is invertible.

A: Given that, V∈Rn,  A∈Mn,n is invertible matrix, i.e  A≠0 we have to show that x,y=Ax.Ay is inner…

Q: Find the gradient field F = Vo for the potential function = 2x³y-4y³x.

A: We have to find out gradient

Q: 4. If A andB are differentiable functions of a scalar u, prove that (А В) — А В du d (a) dA B du du…

A:

Q: Consider the ARMA(3,1) model represented by y, – 0.8y,1-0.25y,-2 +0.2y,-3 = e, -1.2e,1. (a) Is it…

A: Given the ARMA(3,1) model as yt-0.8yt-1-0.25yt-2+0.2yt-3=et-1.2et-1

Q: . Let A- ( 7) B-(: :) and B = %3D 3 For what values of t is 2A+B invertible?

A: The inverse of a matrix A is defined as the matrix A-1, that satisfies the condition AA-1=A-1A=I. If…

Q: 3. Determine whether the functions e and re are linearly independent or dependent by calculating…

A: f= {e^2x , x.e^2x} is Linearly independent if Wronskian is non zero.

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

A:

Q: on vectors for t2 0.

A:

Q: Show that if B is m × n, then BTB is positive semidefinite; and if B is n × n and invertible, then…

A:

Q: Solve for z using cramer's rule.  2x-y-z=4 4x+y+z=2 3y+z=2

A:

Q: Let xe C and ye show that <xiy= y* AX is que inmer product if A is hermitian positive def iwite

A: Let x∈ℂm and y∈ℂm To prove that x,y=y*Ax is an inner product if A is Hermitian positive definite.…

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,…

A:

Q: If Ø : R3 → R3 be defined as Ø (x,y,z) = (2x, 4x –y,2x+ 3yz) then show that Ø is invertible. %3D

A: Given that ϕ:R3→R3 is defined as ϕx,y,z=2x,4x-y,2x+3yz. Now, ϕ is invertible if ϕ is one-one and…

Q: Determine whether the following are linear operatorson Rn×n. L (A) = A − AT

A:

Q: Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or…

A: According to question given :L:R3→R3Lu1u2u3=u1+12u2u3We have to determine whether L is a linear…