Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that (a) U(W) = W;(b) W⊥ is U-invariant.

Q: Let V be the vector space of real 2 × 2 matrices with inner product (A|B) = tr(BtA). Let U be the…

A: Now , V=abcd:a,b,c,d ∈IRbe the Vector spaceof 2×2 real matricesendowed with the inner…

Q: Suppose that V is a vector space over the scalars from R, and T : V → V is a linear transformation…

A:

Q: Prove that if w is a subspace of a finite dimensional vector space V , every linearly independent…

A: To Prove: If W is a subspace of a finite dimensional vector space V, then every linearly independent…

Q: Let V be an infinite dimensional real vector space with a basis e1,e2,e3 . . . and let End(V) denote…

A: Given that  be an infinite dimensional real vector space with basis .Let  denote the set of linear…

Q: Let T be a linear transformation of a vector space V. Prove that{v ∈ V | T(v) = 0}, the kernel of T,…

A: A set S is said to be a subspace of the vector space V if it satisfies the following properties:…

Q: Let T be an orthogonal [unitary] operator on a finite-dimensional real [complex] inner product space…

A: Given: T is an orthogonal operator on V, therefore,

Q: Let T be a linear operator on a vector space V, let v be a nonzero vector in V, and let W be the…

A: An invariant subspace W of T has the property, such that vectors v belongs to W.

Q: Let V be the vector space of real 2 × 2 matrices with inner product  (A|B) = tr(B tA).  Let U be the…

A:

Q: Let V R2[x] be the real vector space of polynomials of degree at most 2. Let {fo, f1, f2} with fo(x)…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Let V be a finite-dimensional vector space and let T: V -> V be linear. Show by Rank-Nullity Theorem…

A:

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: Prove that W₁+W2 is a subspace of V that contains both W₁ and W₂. Prove that any subspace of V that…

A: Definition of W1+W2 is given

Q: Find an example of a unitary operator U on an inner product space and a U-invariant subspace W such…

A: Let V be an infinite dimensional inner product space consisting of finite sequences.Define U:

Q: Consider the vector space V = M2 under the weighted inner product a12 [011 ([an an], [b1 b2]) =…

A:

Q: Let A be a linear transformation from a vector space V of dimension n to a vector space W. Then dim…

A: Given: A:V→W is a linear transformation and dim(V)=n. To show: dim Im(A)+dim Ker(A)=n.

Q: Prove that for every vector v in a vector space V, there is a unique v' in V such that v + v' = 0.

A: We have to prove that for every vector v in a vector space V, there is a unique v' in V such that…

Q: Let V be a C-vector space of dimension n, that is, dim(V) = n. Show that V is an R-vector space of…

A:

Q: Let P, denote the vector space of polynomials of degree < 5 along with the zero polynomial. Define a…

A:

Q: Let V be a vector space of polynomials of degrees ≤ 2 with basis {1, x-1, (x-1)^2) and W be the same…

A:

Q: Let V be the vector space of real 2 × 2 matrices with inner product    (A|B) = tr(BtA).    Let U…

A: Let A∈skewsymm2 and B∈symm2 Then  A,B=trBtA=trAtB , since A,B=B,A=tr-AB=-trBA=-trBtA ,since B∈symm2…

Q: Let V and W be real vector spaces and T be a linear transformation from V into W, that is, T(au +…

A:

Q: Let M2x3(C) be the vector space of 2 x 3 matrices over C, with the usual rules of addition and…

A:

Q: (b) Let V be an n-dimensional F-vector space, and let 0 < k < n. If K is a k- dimensional subspace…

A:

Q: Let T be a linear operator on a vector space V, and let W1,W2, . . . ,Wk be T-invariant subspaces of…

A:

Q: Let T be a linear operator on a finite-dimensional vector space V, where V is a direct sum of…

A:

Q: Let V be the vector space of all 2×2 matrics  over the real field R.  Show that W is not a subspace…

A: Given , V be a vector space of all  2×2 matrices over the real field  ℝ .i.e.    V =    abcd  ;…

Q: Let T : V → V be a linear operator on a finite-dimensional vector space V , such that any subspace W…

A: Let T: V →V be a linear operator on a finite-dimensional vector space V, such that for any subspace…

Q: Let V and W be finite-dimensional vector spaces and T: V →W be linear. (a) Prove that if dim(V)…

A: (a)Let B={v1,…,vn} be a basis for V. Then, C={T(v1),….,T(vn)} spans T(v)