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.

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

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

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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: 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.

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

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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…

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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.