4. Let T: V→ V be a unitary operator on a finite-dimensional complex inner product space V. Prove the following statements. (a) If A E C is an eigenvalue of T, then || = 1. (b) | det (T) = 1.

Q: Let n ≥ 2. Consider the reversing operator R: R" → Rn defined by R(a₁, 02, ...,an-1,An) = (an,…

A: The objective of this question is to find the basis of the eigenspaces E1 and E-1 for the reversing…

Q: 3. Let T: R³ → R³ be the linear operator such that T(e₁) = -e2, T(e₂) = -3, and T(е3) = €₁. (a) Show…

A:

Q: 5. The number 0 can never be an eigenvalue of a linear operator. True or False?

A:

Q: 4. Let T E L(V) where V is a finite-dimensional space. Suppose that A1,..., Am are distinct…

A: It is given that

Q: 1 2 2 and define Q : R² → >R by Q(x) = x" Ax. (a) Find the characteristic polynomial of A. (b) Find…

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

Q: 6. Let T: P₂ → P₂ be the linear operator defined as T (p(x)) = p(3x), and let B = {1, x,x²} be the…

A:

Q: - Let T€ B(H). (a) Show that ker T'T=ker T. (b) Show that TT and TT have the same eigenvalues. (c)…

A:

Q: 10. Let P denote the vector space of polynomials of degree at most k. Let T: P3 → P2 be the function…

A:

Q: (2) Suppose that V is a finite dimensional vector space over R. Show that if dim(V) is odd, then…

A: For the solution follow the next step.

Q: in ExerCIses 11. : Z- = 6. Zs such that (1) 12 : Z Z such that o(1)312. Z x Z Z where (1,0) = 3 and…

A:

Q: Show that for any λ E F and any positive integers 1 ≤ d <≤m, there exists a linear operator T: V → V…

A:

Q: True or False: There exists a linear operator T E L(C°) whose minimal polyno- mial is (z – 2)2 and…

A:

Q: (b) Let v, 2,... , vn be a basis of a vector space V. Suppose f: V -> V is a lincar map so that Vi…

A: To work out the required properties of the linear operator f.

Q: [1 2 8. Let A = and define Q : R² → R by Q(x) = x"Ax. 2 1 (a) Find the characteristic polynomial of…

A:

Q: 8. Let P3(C) be a vector space over C and let D: P3(C) → P3(C) be the differential operator defined…

A: D: P3ℂ→P3ℂ be the differential operatorDpt=dpdtrelative to the basis 1, t, t2, t3

Q: THEOREM 9. Let K be a self-adjoint operator on a Hilbert space H. All eigenvalues of K are real. rl.

A:

Q: defined by Let n ≥2. Consider the reversing operator R:R" R(a1, a2,,an-1, an) = (an, an-1,..., a2,…

A: The objective of the question is to find the basis of the eigenspaces E₁ and E₋₁ for the reversing…

Q: Let k be the k-th coordinate functional corresponding to the natural basis of 1, that is k(x)=x(k)…

A: We have to show that φk is linear, we need to demonstrate that it satisfies the properties of…

Q: Let T : R” → R” be defined by T(x) = A() where A is an invertible ʼn × ʼn matrix. Then I' is an…

A:

Q: Just

A: We use definition of linear operator to find matrix matrix representation with respect to basis.

Q: 5. Let T: V→ V be a linear operator on a finite-dimensional complex inner product space V. Show that…

A: Useful result:

Q: Let D : Pn(R) → Pn(R) be the differential operator defined by D(f) = f'. Show that the matrix…

A: I have used the fact that non zero nilpotent matrix is not Diagonalisable

Q: Let T: VV be a linear operator on a finite-dimensional F-vector space V, and let f(t) be the…

A:

Q: 2. Linear Maps (a) Let Pn denote the space of polynomial of degree at most n with real coefficients.…

A: We have to find the matrix representative of differential operators using standard monomial basis.

Q: 15. Assume A and B are unitary matrices (A) The determinant of A can be -1 (B) (AB)T= A-¹B-1 (C) A…

A:

Q: 3) Suppose T E L(V) and T² = I and -1 is not an eigenvalue of T. Prove that T = 1. %3D

A:

Q: defined by Let n ≥2. Consider the reversing operator R:R" R(a1, a2,,an-1, an) = (an, an-1,..., a2,…

A: The objective of the question is to find the basis of the eigenspaces E₁ and E-1 for the reversing…

Q: A linear operator T: V→V is called nilpotent if TP = 0 for certain positive integer p. Prove that if…

A: Given that T:V→V is linear operator. A linear operator T:V→V is called nilpotent if Tp=0 for certain…

Q: 2. Let T: R³ R³ be the operator given by 1 1 T(v) = 9). 1 Determine whether T is decomposable or…

A: Consider the given operator, Tv=-1100-1100-1v Here we need to determine that whether T is…

Q: 2. Let A be an m x n matrix over complex field. (a) Prove that if x = N(L), then Ax is in both R(L)…

A: note : Since you have posted question with multiple sub parts, we will provide the solution only to…

Q: Let T∶V → V be an invertible operator and let λ be an eigenvalue of T. (1) Prove that λ ≠ 0 (2)…

A: be an invertible operator and is an eigen value of

Q: Let A be a real symmetric matrix. Show that all eigenvalues of A are nonneg- ative if and only if A…

A: