Consider the vector space V = C([0, 1]) of complex-valued functions that are continuous on [0, 1]. Show that for every alpha α ∈ C and f ∈ C([0,1]), we have  (a) ⟨f − αg, f − αg⟩ = ⟨f, f⟩ − 2Re(conjugate of α⟨f, g⟩) + |α|2⟨g, g⟩ ≥ 0. For g ≠ 0, set α = ⟨f, g⟩/⟨g, g⟩, which is defined since ⟨g, g⟩ = ||g||22  ≠ 0. With this value of α, prove the Cauchy-Schwartz inequality for L2: |⟨f, g⟩| ≤ ||f||L2 ||g||L2.

Q: 6. Suppose V is a complex vector space with dimension n and T E L(V) is invertible. Let p and q be…

A: In the question it is asked to prove p(0)q(z)=znp(1z).

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Q: Let V be the subspace of the vector space of all real-valued functions that is spanned by the setS…

A: If V is the subspace of the vector space of all real-valued functions that is spanned by the set S…

Q: 5. Let F(x, y) = ei + xej be defined on a smooth curve C given by the vec function r = ti+t2j for 0…

A: A detailed solution is given below

Q: Consider the vector space P, of all polynomials of degree = a, a,+b, b2 , then the inner-product of…

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Q: 7. Let C[-7,7] be the vector space of continuous function over [-n, 1] with an inner product 1 (5,9)…

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Q: If V(F) is a vector space of polynomials in t of degree at most 3 and D be the differentiation…

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Q: 4. Let F be a smooth vector field. Show that the flux of F leaving an infinitesimal cube of volume…

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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: Use the inner product (f,9) = f(z)g(x) dz in the vector space C"0, 1] of continuous functions on the…

A: It is given that fx=-5x2-3 and gx=9x-7. We have to evaluate f, g, f, g and αf, g where αf, g is the…

Q: Show that the field R(i) as a vector space over R has a basis { 1 , i } and is isomorphic to  R_2,…

A: The field ℝ (i) is defined as ℝ (i) = {a + bi| a, b ϵ ℝ}.That implies, ℝ (i) = {a × 1+ b × i | a, b…

Q: (d) Consider the vector space C0, 1] of all continuously differentiable functions defined on closed…

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Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

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Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos/3…

A: Solution: Here the sequence is xnt=4cosn3t5+cosnt and a=-5π,  b=-4π

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Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

A: Given : V is any normed vector space and x is any non zero vector and α>0. To find :  A real…

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A: A linear operator is called self-adjoint if T=T*.

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Q: 11. Let (V.||-|) be a normed vector space. Suppose e : V - R has the property that e(r+y) =…

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Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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Q: Let F(x, y) = (4 cos(y), 2 sin(y)). Compute the flux $ F - n ds of F across the boundary of the…

A: By using Green Theorem, ∮F.n ds…

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A: Given, A=2112, x=11

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Q: On a vector space W of all continuous functions from R to R, define I, : W → W as I,[f(t)]= J', f(t)…

A: Yes  given function is linear transformation

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Q: Let F be the vector space of infinitely differentiable functions from R to R. Consider the subspace:…

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Q: 5. Let V be a vector space, and let x1, X2,.. . , Xk E V. 5(a) Prove that if either x1 = 0 or x;+1 E…

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Q: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: Use the inner product (f, 9) = ". f(x)g(x) dx in the vector space Co[0, 1] of continuous functions…

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Q: /- X be the vector Space of all Complex 2x2 and de fine T: x →x by Tx = bx, where Let matrices bEX…

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Q: a) T: M, → R, where M,n is the vector space of n by n matrices over a field, and T (A) = det(A). b)…

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Q: 2. Let w be a positive continuous function on [0, 1], n be a positive integer, and Pn be the vector…

A: a)Note that the definition of inner product here is completely irrelevant, for any inner product on…

Q: Let f: C → C be differentiable at zo. Considering f as a map R? → R², that f'(zo) = kR where k is a…

A: (a)

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Q: let T be a linear operator on a finite- dimensional vector space V, and let f(t) be the…

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Q: 1) Verify that d/dx(fg) is a vector space over R, or explain why it is not.

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Q: a) T: M, → R, where M, is the vector space ofn by n matrices over a field, and T (A) = det(A). b)…

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Q: Determine a scalar r such that Ax = rx, where

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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.