(a) Let A and B be square matrices that are unitarily equivalent. Prove that ||A||= ||B||. (b) Let T be a linear operator on a finite-dimensional inner product space V. Define Prove that ||T||= || [T]β||, where β is any orthonormal basis for V. (c) Let V be an infinite-dimensional inner product space with an orthonormal basis {v1, v2, . . .}. Let T be the linear operator on V such that T(vk) = kvk. Prove that ||T|| (defined in (b)) does not exist.

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

A:

Q: Prove the spectral theorem for Hermitian operators, H, in a finite dimensional inner product space…

A: Spectral theorem for Hermitian operators H in a finite dimensional inner product space V:…

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

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Q: Let V be the vector space of symmetric 2 x 2 matrices and W be the subepace w - q W- span Find a…

A: First we calculate the basis of V.

Q: 2. Let V be a vector space of dimension n and let B = {v1,..., Un} be a basis of V. (a) Prove that…

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Q: Let V be an n-dimensional vector space with basis B = {v1, .•. , vn}. Let P be an invertible n X n…

A: Given: V is an n-dimensional vector space with basis B={v1,....,vn}.To prove: C={u1,....,un} is a…

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: 3. Prove that, for a given n, all n-dimensional vector spaces with real scalars are isomorphic.

A: A vector space V will be isomorphic to other vector space W if there exists a linear transformation…

Q: Prove that the spectral norm of a symmetric matrix is the same as its spectral radius.

A:

Q: If the percent decrease is 25%, what is the common ratio?

A: Common Ratio The common ratio is defined as the amount between each number in a geometric sequence.…

Q: Prove that the adjoint operator is linear and bounded

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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: Let T : V → V be a linear operator over a finite dimensional vector space V . Prove that dim(ker(T))…

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Q: Let B = (v₁.....V₁) be a linearly independent set in R". Explain why B must be a basis for R".

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Q: Show M={A in M2,2| |A| =0} is not a vector space under the usual addition and scalar multiplication…

A: We will show M={A in M2,2| |A| =0} Is not vector space by exhibiting two elements A,B belongs M but…

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

A:

Q: Every finitely generated vector space has a finite basis

A: Every finitely generated vector space has a finite basis.

Q: Please see image

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Q: T= is a vector space under standard matrix addition and scalar multiplication

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: Solve using the method of undetermined coefficients: y" + 3y" – 10y' = 2 + xe* %3D

A: Given: y''' + 3y'' - 10y' = 2 + xex

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: Let V be an inner product space. Show that ||cu|| |c| ||u|| for any vector u and any scalar c.

A: Given:          V is an inner product space.   To show :            cu = c u for any vector u and…

Q: Let be the vector space of polynomials of degree less than or equal to one with real coefficients…

A: The given vector space is the set of all polynomials of degree less than or equal to one with real…

Q: Show that a bounded linear operator A on complex Hilbert space H is self adjoint if and only if is…

A:

Q: Let X1=[1,0,2,-1] , X2=[-1,1,0,1] , and X3=[2,0,0,-2] and let W= Span{X1, X2 , X3}. Find an…

A: Concept: According to the Gram-Schmidt process, uk→=vk→-∑j=1k-1projuj→(vk→)uk​​=vk​​a vector…

Q: (a) Given a finite-dimensional inner product space V and a linear operator T:V V, explain how w…

A:

Q: Show that it is impossible to find a basis for the vector space ofn x n (n > 1) matrices such…

A: Basis: A basis is the set of vectors of the vector space V which is linearly independent and the set…

Q: Determine whether each of the following is a basis of the vector space P,(1): (a) {1, 1+1, 1+r+²,…

A: Basis of a vector space : A subset S of a vector space V(F) is said to be a basis of V(F) if (1) S…

Q: Consider the vector space V over the real numbers, consisting of all polynomials of degree less than…

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Q: Every finitely generated vector space has a finite basis.

A: Every finitely generated vector space has a finite basis.

Q: Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with the…

A:

Q: In each case, determine whether V is a vector space. If it is not a vector space, explain why not.…

A: We know that a non empty set V is a vector space over a filed F if V is an abelian group under…

Q: Prove that any complex inner-product (,)v on a complex vector space V, there is a basis U = {₁,...,…

A: We need to show that, x,yV=xU, yUH In other words for any finite dimensional inner-product space,…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.