Let T be a linear operator on a real inner product space V, and define H: V ×V →R by H(x, y) = <x,T(y)>for all x, y ∈V. (a) Prove that H is a bilinear form. (b) Prove that H is symmetric if and only if T is self-adjoint. (c) What properties must T have for H to be an inner product on V? (d) Explain why H may fail to be a bilinear form if V is a complex inner product space.

Q: Let V be the set of all ordered pairs of real numbers, and consider the following (y1, Y2) in V and…

A: X  = (x1,x2) Y  =(y1,y2) v,k from R

Q: Is the set V = {(x, y) x >0} where (x1, y₁) + (x2, Y₂) = (x₁x2, y₁ + y₂) and c (x, y) = (x,cy) a…

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Q: Let x and a be vectors in R2 Define the L: R2→ R2 by L(x) = x +a Is this function a linear operator?…

A: Linear operator should satisfy L:Rn→Rn  a) Lu+v=Lu+Lv         u,v∈Rn b)Lcu)=cL(u

Q: Let T: P₂ P₂ be a mapping defined by T(a + a₁x + a₂x²) = ao + a₁ (x + 1) + a₂(x + 1)². a) Determine…

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Q: Let P₂ be the vector space of polynomials of degree at most 2. Let T: P₂ → P₂ by T (p(x)) = p(x) + x…

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Q: Let S be the space of all complex vectors (x, y, z) such that y = z². Determine whether or not S is…

A: In the given problem we verify the properties of vector addition

Q: be the vector space of all polynomials of degree 3 or less in the variable z. Let PI(T) 2+x+1²,…

A: In this question, we check the given set forms a basis or not. Then we use elementary transformation…

Q: .. Let V R[r] be the vector space (over R) of all polynomials in a with real coefficients. Define…

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Q: c. Show that V, with respect to these operations of addition and scalar multiplication, is not a…

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Q: 1. Let P(R) denote the vector space of polynomials with coefficients in R. Define a function T: P(R)…

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Q: Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive…

A: Below given clear explanation

Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: Show that if V is a real vector space and av=0, then a=0 or v=0

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Q: Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map…

A: Follow the steps.

Q: V = set of all pairs of real numbers. (x,y) Q3 Prove that such as 2 (x, y) + (x²₁ y ₁ ) = (x + x² +…

A: Given:V= the set of all pairs of real numbers x,y.The operations are given…

Q: Suppose W1, ..., Wm are vector spaces. Prove that L(V, W1 × • × Wm) and L(V, W1) × ·…× L(V, Wm) are…

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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: 3. Let V denote the vector space of all functions f : R → R, equipped with addition + : V x V → V…

A: (a) Given that V is a vector space of all real functions defined on real number. W= {It is given…

Q: Let T be a linear operator on a vector space V, and let W be a T-invariant subspace of V. Prove that…

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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.