Let V be an inner product space, and let T and U be linear operators on V. Then (a) (T + U)∗= T∗+ U∗ (b) (cT)= cT∗ for any c ∈F (c) (TU)∗= U∗T∗ (d) T∗∗= T (e) I∗= I.

Q: Let T be a linear operator on an inner product space V. If (x), y>= 0 for all x, y ∈V, prove that…

A: Consider a basis for V as follows.

Q: (Proving Linear Algebra)

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Q: (d) If S,T are symmetric linear operators on an inner product space V, then their composition ST is…

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Q: Let V be an inner product space, and suppose that x and y are orthogonal elements of V. Prove that…

A: Given:

Q: Let X be a normed linear space and T: X→ X and S: X → X be two bounded linear operators, then SoT…

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Q: Let U be a linear operator on a finite-dimensional inner product space V. If ||U(x)||= ||x||for all…

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Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

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Q: Give an example of a linear operator T on R2 and an ordered basis for R2 that provides a…

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Q: Let V be a finite-dimensional vector space and let T be a linear operator on V . Suppose that…

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Q: .

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Q: Let T be a linear operator on an inner product space V, and supposethat IIT(x)ll = llxll for all x .…

A: Let us consider T be a linear operator on an inner product space V, and suppose that IIT(x)ll =…

Q: Let B be an invertible matrix in M (R). Define T: M (R) → M_ (R) by nxn T(A) = B AB. Prove that T is…

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Q: Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on…

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Q: Prove that if T is a unitary operator on a finite-dimensional inner product space, then T has a…

A: It is given that T is a unitary operator on a finite dimensional inner product space.

Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: 3.) Let V be a finite-dimensional vector space. Suppose T : V → V is a linear transformation. how…

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Q: Let E be a vector space. If E= U+V and E=U+W, then there is an isomorphism f: V--W between

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Q: Let V be an n-dimensional vector space with an ordered basis β. Define T: V →Fn by T(x) = [x]β.…

A: Consider the provided transformation:Define T: V →Fn by T(x) = [x]βNow, let β ={v1, v2,……,vn} be the…

Q: Show that {t, e'}, which is a subset of F, the vector space containing all real-valued functions, is…

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Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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

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Q: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

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Q: Let V be the set of 2 × 2 matrices with entries from R, that is, V ={ [ a b c d ] | a, b, c, d ∈ R }…

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Q: A D,C are three linear transformations on a vector space V(F) such that AB = CA = I, then prove that…

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Q: Let H, and H, be Hilbert spaces and T: H H, a bounded linear operator. If M,cH, and M2 c H2 are such…

A: Given H1 and H2 be Hilbert space and T:H1→H2 a bounded linear operator. If M1⊂H1 and M2⊂H2 are such…

Q: Let V, W be vector spaces over F and let T:V → W be a linear map. Prove that if T is injective, then…

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Q: Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a bounded linear operator. Let M1 = N(T)…

A: Given:- Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a boundedlinear operator. Let M1…

Q: Let V be an inner product space over R. Let u1, U2, Uz, U4 E V such that (u;, u;) < 0, i, j = 1, 2,…

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Q: Let V be a finite dimensional vector space, with S : V → V a linear operator, and ⟨·, ·⟩ an inner…

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Q: Let T be a diagonalizable linear operator on   V -finite-dimensional, and let m be any positive…

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Q: • Exercise 3 Prove that if S and T are both bounded linear operators from the normed space X to the…

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Q: Consider an n-dimensional vector space (V, R), that is, there are up to n eleme.ts of V that can…

A: We will first assume that vectors in W are linear dependent over R and prove that aour assumption…

Q: Let V be a finite-dimensional inner product space, and let E be an idempotent linear operator on V.…

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Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

Q: Let V and W be finite-dimensional vector spaces, and let T: V → W be a linear transformation.…

A: Let T: V-&gt;W be a linear transformation. Β is a basis for V.It is required to prove that T is an…

Q: Let V be a K-vector space and U1, U2, U3 three sub-vector spaces of V. Show that: dim(U1) + dim(U2)…

A: first of all we must know about the dimension of the vector space.what is the dimension of the…

Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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Q: If B is a vector in an inner product space, if T is o linear transformation on that space, ond if…

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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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Q: Let T be a linear operator on a real vector space V , and define f : V × V → R by f(x, y) = (x, T y)…

A: As per company guidelines we are allowed to solved maximum three sub-parts so i am solving first…

Q: Let V and W be vector spaces with ordered bases E and F, respectively. If L : V → W is a linear…

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Q: Let T be a linear operator on an inner product space V. Let U1= T + T* and U2 = TT*. Prove that U1 =…

A: If T is an operator in an inner product space V over the field F together with an inner product,…

Q: Let T be a linear operator on a real inner product space V, and define H: V ×V →R by H(x, y) =…

A: Given: T is a linear operator on V.(a) To show: H is a bilinear form.H is defined as,

Q: Define A : L3/2[–1, 1] → C by Prove that A is a bounded linear map and compute its norm.

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Q: Let T: U→V be a linear transformation from a vector space U (F) into a vector space V(F) and U(F) be…

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Q: Let R[r], be a linear space with all polynomials p(x) = az²+bx+c (a,b,c ER) whose degree is no more…

A: We have to find the matrix A representing L with respect to the given basis. Then we can proceed…

Q: 7. Let V be a vector space and l: V → R be a linear map. If z E V is not in the nullspace of l, show…

A: Given: 7) V is a vector space and l:V→ℝ is a linear map. To show: If z∈V is not in the null space of…

Q: Give an example of a linear operator T on an inner product space V such that N(T) ≠ N(T∗).

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Q: Let T be a linear operator on an inner product space V, and suppose that ||T(x)||= ||x|| for all x.…

A: According to property of norm the value of ||x||=0 iff x=0.

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.