14. Let T be a linear operator on a finite-dimensional inner product space V. Then T is self-adjoint if and only if its matrix in every orthonormal basis is a self-adjoint matrix.

Q: Let V, W be finite-dimensional vector spaes over C. (a) If T ≤ L(V, W), prove the Rank-Nullity…

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Q: There exists an ordered basis for V such that every operator

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Q: Let B = {B]. A} mation x→→ Ax. =[%] be a basis for R2 and A = Find the B-matrix for the transfor-

A: Given that The transformation is

Q: (a) Construct the 3 × 3 matrices A₁ and A₂ associated to the linear transfor- mations L₁ and L2,…

A: (a)Two Linear transformations are mentioned:.Thus the matrix for the transformation  is given by and…

Q: Q4(a) Define the null space of an m x n matrix A and prove that it is a subspace of R"

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Q: that if all the components of a tensor in one co ordinate system are zero

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Q: 2) Let v be the rector space of matrices defined in R. 4 A=[ ethgiLED Find the coordmates of the…

A: Given that, A = 4-11-11-7 Let X=1-2-21, 2113, 4-1-15 Coordinate representation relative to the basis…

Q: 1 2 0 0 -5 0 0 1 0 -4 0 0 0 1 2 Lo 0 0 0 0J

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Q: Let R[x] be a linear space with all polynomials p(x) = ax²+bx+c (a, b, c E R) whose degree is no…

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Q: values of (1, Ar) on the manifold where (1, B1): Let A and B be Hermitian operators on a real…

A: As per the question we are given an invariant manifold on the Hilbert space defined by : ⟨x,Bx⟩ = 1…

Q: Show that any isometry of R^n which fixes the origin is linear, and hence must be of the form f(x) =…

A: Introduction A square matrix is said to be an orthogonal matrix if…

Q: Justify if it is true: The set of forms a space vector. eigenvectors of an endomorphism T,

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Q: ()-() - y +22) 2y Sei f : R → R*, (y 2x + 4z a) Geben Sie die darstellende Matrix M(f) von f bzgl.…

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Q: B = (v₁, V₂, (The ia Let V be an inner product space (over C or R), with an orthonormal basis Vn)…

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Q: 33. T(x1,x2,X3) = (3x1, x2, X1 – x2)

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Q: .

A: Given two bases of R2:

Q: Let L: R2 R2 be defined by L 4*]-[***] represents the coordinates of v in the standard ordered…

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Q: a) Give an example of subspaces U = span(f1, f2, f3) and V = span(g1,92; g3) of RaJa] such that…

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Q: Characteristic spaces of a normal operator are pair-wise orthogonal.

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Q: Exhibit a basis for T2, the vector space of all 2 x 2 matrices of real numbers with zero trace.

A: Let V be a vector space over a field F. A set S of vectors in V is said to be a basis of V if  (i) S…

Q: 8. = - Prove that the set is a basis for the vector space of 2x2 matrices

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Q: Let Riz be a linear space of all polynomials p(x) = ar²+bx+c (a,b,c & R) whose degree is no more…

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Q: Let Pn = {anxn + an-1xn-1 + . .  + a1x + a0 | each ai is a realnumber}. Is it possible to have a…

A: The polynomial space is defined as follows: Pn=anxn+an-1xn-1+⋯+a1x+a0  |ai∈ℝ Check whether is it…

Q: 5. Let M3 - {{: ahader} M22 = {" a la,b,c, de R Show that M22 with the usual matrix addition and…

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Q: 2) Let H be a Hilbert space, express H as a direct sum of any two of its subspaces. Prove that the…

A: 2)It is given that H is a Hilbert space. Let, M and M⊥ be closed subspace of H.

Q: Let Z be the linear operator on R² defined by L( i). Find the matrix A representing L with respect…

A: Basis: A set of vectors which is linearly independent and spans the whole vector space is calledthe…

Q: Verify by finding the matrix of S 0 T (a) by direct substitution and (b) by matrix multiplication of…

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Q: 5. P2 denotes the set of all polynomials of degree ≤ 2, in a single variable x, with real number…

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Q: 2. Consider the basis * B = (1+*+**, x +r, 2). %3D for P2. Let p(x) = -1 + x + 2r?. Then [p(x)]R =…

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Q: Let operator be T:R³ → R³ with correspondence rule T(x, y, z) = (y, x + z, 0) a) Determine the…

A: Let the operator T: ℝ3→ℝ3 such that T(x, y, z) = (y, x+z, 0) a) Characteristic values of T b)…

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Q: im(T1 + 14T) im(14T† + T2).

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A: The null space of T is the collection of all polynomials p(x) in the domain of T such that T(p(x)) =…

Q: Let Mı5(IR) = {A/A and 15th order square matrix with real elements} and consider the linear…

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Q: ༤ས་ Let A = = [3 0 0 -3 -2 -1 A basis for the null space of A is { vector

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Q: Let V be an F-vector space and A, B, C are subspaces of V. (a) Show that the equation An (B+C) =…

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Q: Let A be a bounded linear operator on a Hilbert space H and ||A(x)||=||x|| ¥xe H. Show that A is…

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