(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

A:

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: Let A = {a1, a2, a3} be an ordered basis for the real vector space V. (a) Find the coordinate matrix…

A: Let A=α1, α2, α3 be an ordered basis for the real vector space V. For (a), We need to find the…

Q: Find the general solution of the differential equation: y+2y = test Use lower case c for the…

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

A:

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: 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: 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 V be a vector space of dimension 4, and let W be a vector space of dimension 2. Let L:VW be a…

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

A:

Q: Let V be the vector space of functions spanned by Nπ B = {b₁ = 5 sin x + 7 cos x, b₂ = 3 sin x + 9…

A:

Q: 3. Let P2 be the vector space of polynomials of degree at most 2. Define a map D: P2 → P2, it sends…

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Q: 3. Let V be the vector space of polynomials with coefficients from R. a) Show that the following…

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Q: Let T be a linear operator on a two-dimensional vector space V and suppose that T≠c| for any scalar…

A: Let T be a linear operator on a two-dimensional vector space V.

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

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

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Q: Consider the vector space generated by [0, 1, –1, –1], [1,0, 2, 4], [1, 2, –4, 0]. Find an…

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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 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: 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: 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: If V is a finite-dimensional vector space, then a bilinear form f on Vis symmetric if and only if…

A:

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

A:

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,…

Q: Let P3[x] be the vector space of all polynomial of order at most 3. Suppose that B = { 1, (x-1),…

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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…

A:

Q: Find a basis for the span of {(1, 1, 0, 1), (0, 1, 1,0), (1, 0, 1, 1), (1,0, 1,0)} in the vector…

A: Observe the fact  that  . Then, Now, Thus, the vector  will  not  be a  basis  vector.Let us…

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.