7. Let be the operator on P[r]3 defined by L(p(x)) = xp'(x) +p"(x) (1) Find the matrix A representing with respect to [1, x, x²]. (2) Find the matrix B representing with respect to [1, x, 1 + x²]. (3) Find the matrix S such that B = S-¹AS.

Q: Prove that Ją (2) = =- J¼ (2) – J_4 (=)

A:

Q: Evaluate the commutators of the operators (a) d/dx and 1/x, (b)d/dx and x2.

A: Given:  a)  ddx and 1x b) ddxand x2 To Find: Commutator of the operators

Q: dz (b) Determine the by using the Chain rule with the following information: ds z = cos (rªy^) – sin…

A:

Q: Suppose you know that f:R? → R³ is a linear function, and that (1)- 5 and 2 (a) (6 points) What is ƒ…

A:

Q: If A and B are orthogonal, is A+B also orthogonal?  Can you demonstrate Algebraically using A + B =…

A: Solution: Given-   A and B are Orthogonal matricesTo show:  (A+B)  is OrthogonalSince, A and B are…

Q: Let A be an n xna diagonalizable matrix. Prove that the matrix exponential satisfies det(e^) =…

A: According to the given information, Let A be an n × n diagonalizable matrix. It is required to prove…

Q: Find the matrix A of the linear transformation from R² to R³ given by 1 0 -8 x2. -4 x1 + 2 -6 T…

A:

Q: Write a'= -3x+y+ln(t), y'= t'a in matrix form as 18 31 and f(t) is 181 [*] -= P(1) [*] + f(t) where…

A:

Q: 3. If B= (-3, 2), (4, -2) and B' = (-1,2), (2,-2) are two bases for R² (a) Find the transition…

A:

Q: Consider i) Show that B and B' are bases for M22- ii) Find the coordinate matrices of A =; relative…

A: a. The dimension of M2, 2 is 4. The dimension of B is 4 and the dimension of B' is 4. All elements…

Q: Determine the kernel and the range of the linear operator on L(x) = (x1 + x3, X2, 0)

A:

Q: Let 8 = {(1, 3), (-2,-2)} and 8' = {(-12, 0), (-4, 4)} be bases for R² 4 2 A - [83] be the matrix…

A: Note: Since you have posted a question with multiple sub-parts, we will solve the first three…

Q: 4) Find the adjoint of the operator Lu = u" + 4u'- 3u, where u'(0) = -4u(0), u(1) = -4u(1).

A: The objective is to find the adjoint of the operator, Lu=u''+4u'-3u where u'(0)=-4u(0), u'(1)=-4u(1)…

Q: 5. If xy + 2xu + 3xv + uv-| = 0, 2xy + 3yu – 2xv + 2uv + 2 = 0, ſind the Jacobian matrix [ ].

A: Let, F(u,v,x,y)=xy+2xv+3xv+uv-1Gu,v,x,y=2xy+3yu-2xv+2uv+2 By the chain rule,…

Q: 5) Find dx/dt of Matrix: X 5e-1 2e-¹ -7e-t

A: We know , d/dx(e-t) = -e-t   Given X = 5e-t2e-t-7e-t

Q: find linear systems (matrix) x'(t) =[ 11 - 15 6 - 7 ] x(t) classify its origin

A:

Q: Let B = and B' = Find the change of base matrix from B' to B and from B to B'

A:

Q: Orthogonalize {1, t, t²} in P₂ (R) with 1 (f, 9) = ["ª, f(x)g(x) dx. -1 Then normalize each one so…

A:

Q: If A is an adjoint operator on H=K2 and  for x=(x(1), x(2)), we define A(x)=(x(1)-x(2), x(1)+x(2)),…

A:

Q: linearise R = AT + BT

A: To linearise the below equation. R=AT+BT2

Q: Question 1 (Mapped to ILO1) Direction: Perform the indicated operations for the given Matrices and…

A:

Q: Find e^t where A is the matrix A - -4 1 -1 -2 eAt +

A: Given that A=−41−1−2

Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: Need help please address all the questions, please type the answer by computer clearly  a p= b…

A: Two given bases for ℝ2 are B=1,2,-1,-1B'=-4,1,0,2 Also A=20-11 be the matrix for T:ℝ2→ℝ2 relative to…

Q: 4) Determine whether thé givel a) f, (z) = cos 2r, 5(2) = 1, 1,(2)= cos' z b) f(2) = z. 1,(2)= z-1,…

A: Given:- a) f1(x)=cos2X, f2(x)=1, f3(x)=cos2x b) f1(x)=x , f2(x)=x-1 , f3(x)=x+3 c) f1(x)=ex+2…

Q: : {1+x,1 − x²,1+x+ x² } and B' = {2 − x, 1 − x+x²,3+2x}. 3.) Find the transition matrix from B to…

A: