Consider the vector space of odd real periodic function f: R these function are in the region-LSxSL The base for odd periodic function sin () - Rwhere domain of Prove this basis is orthonormal. for any integer n

Q: 1. Let n e N and let V be the set of polynomials in the indeterminate x with coefficients in R of…

A: According to our guidelines we are only supposed to answer the first three subparts of the question.…

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

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Q: Let P be the set of all degree <4 polynomials in one variable x with real coefficients. Let Q be the…

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Q: Find the dimension of the vector space V and give a basis for V. (Enter your answers as a…

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Q: Consider inner-product = :| f{x)g(x) dx in vector space C[- 17, n], the Fourier expansion of f(x) =…

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Q: (x|y) (x]x;)(y|X;).

A: Let, x,y∈V. Given that B=x1,x2,...xn is the orthonormal basis. So, we can write as,…

Q: (a) Let V = F[x] be the vector space of polynomials over a field F. Show that the set S = {1, x,…

A: (a) let V=Fx be the vector space of polynomials over a field F. show that the set S=1,x,x2.x3,... is…

Q: Prove that in the set of real-valued functions {sin t, sin 2t, sin t cos t} is not a basis for the…

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Q: 5. Let V be the vector space of polynomials in t with inner product defined by (f, g) = [ f(t)g(t)…

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Q: (b) Find an isomorphism between the vector space of all polynomials of degree at most 3 such that…

A: As per your query the parts (b) and (c) are done in the following steps.

Q: Q.2) Define a vector find in 2-space with an yi + xj example. Graph of the vector field F(x, y)

A: A vector field on 2 dimensional space is a function that assigns to each of the point (x, y) in…

Q: Suppose that V is a finite dimensional real inner product space with orthonormal basis V1, V2, V3 .…

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Q: (d) Show that the polynomials P₁ = 3 + 2x + x², P₂ = 2x+3x², p3= −2+x=2x² form a basis of the vector…

A: Using 1+2x-x^2=ap1+bp2+cp3 find a b c

Q: show that vectors Centywhere en I's the sequence whose nth term i's 1 and all o ther terms are Zero…

A: Given: The vectors en where en is the sequence whose nth term is 1 and all other terms are zero.…

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

A: Given D is differentiation transformation. The basis is 1, t, t2, 1+t3. T(1)=ddt1=0…

Q: (3) Let T be a linear operator on a vector space V of dimension n. Suppose v1,...,Vn is a basis of V…

A: We are given that T is a linear operator on a vector space V of dimension n. And v1, v2, v3, ... ,…

Q: 2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the…

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Q: Consider vector space V over field R. Consider linearly independent sequence (v1, v2, . . . , vn)…

A: Option (b) is correct.

Q: Let e1, e2,..., en be an orthonormal list of vectors in an inner product space V over F. For any…

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Q: Linear Algebra

A: We are given that W = span{x, y}.  Here, we need to construct an orthonormal basis for W. In other…

Q: Consider polynomials with degree less than or equal to n. We can think that each polynomial…

A: First find map on polynomial then equivalent to tuples

Q: (p, q) pq dx use Gram-Schmidt to find an orthonormal basis for P2[c], the space of all polynomials…

A: By using gram Schmidt process we solve the problem.

Q: Let v=(-2,3,0,6), find all scalars k such that |kv|=5 LUTION

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Q: Consider inner-product =| f(x) g(x) dx in vector space C[- n, 7], the Fourier expansion of f(x) = x…

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Q: Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients,…

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Q: Let W be the subspace of the space of all continuous real- valued functions spanned by {cos2 t,sin2…

A: let U= cos2t,sin2t,cos2t and W be the subspace of all continuous real-valued functions spanned by…

Q: Let V be a finite dimensional vector space over R, with a positive definite scalar product. Let…

A: 8) In this question, we need to prove  &lt;v,w&gt;=X.Y

Q: Let B = {1, x, x², x³, x4}, which is a basis for the vector space of polyno- mials W = P4(R). Let V…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Give a basis for the indicated vector space over the field. Indicated Vector Space:  ℚ(i) over ℚ

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Q: 2. Let T be the normal operator on the finite-dimensional complex inner product space V. Then T is…

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Q: 8. Let P[0, 1] denote the complex vector space of all complex valued polynomials defined on [0, 1].…

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Q: Let n € N and let V be the set of polynomials in the indeterminate x with coefficients in R of…

A: We will be solving part (b), (c), and (d) as mentioned. Given n∈ℕ. Now define:…

Q: Let T be a linear operator on a finite-dimensional vector space V such that the characteristic…

A: Let T be a linear operator on a finite-dimensional vector space V such that the characteristic…

Q: Let V be the set of polynomials with real coefficients having degree at most 2. Prove that V is a…

A: let V be the set of polynomials with real coefficients having degree at most 2. to prove- (i) V is a…

Q: Let e1, e2,..., en be an orthonormal list of vectors in an inner product space V over F. For any…

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Q: (a) Show that the list {p o(x), p 1(x), p 2(x), ...,p n(x),...,} where po(x) = 1, p1(x) = 1+x, P;(x)…

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Q: )Let P3 be the vector space of all polynomials of degree 3 or less in the variable z. Let Pi (z)…

A: Given that P3 is the vector space of all polynomials in variable x of degree 3 or less.We have to…

Q: Let Pn be the vector space of all polynomials of degree n or less in the variable æ. Let D² : P4 →…

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

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Q: Consider inner-product = | f(x)g(x) dx in vector space C[- n, T], the Fourier expansion of f(x) = x…

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Q: Denote three scalars A = eTE-le, B = µTE-le and C = µTE-'µ. 1. Prove that AC – B² > 0.

A: We are given three scalers A,B,C in terms of weighted matrix inner-products, and we have to prove…

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: Find a basis {p(x), q(x)} for the vector space {f(x) E P2[x] | f' (9) = f(1)} where P2[x] is the…

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Q: 13) Let V be a complex vector space. Suppose that T is a self-adjoint operator on a inner-product…

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Q: Let V be the set of sequences of real numbers. For {an},{bn}belongs to V and any real number t,…

A: Given  V is the set of sequences of real numbers. So we can write V as:

Q: Consider vector space V, with finite dimension n over field F and two bases B and C of Vn. Then…

A: Fourth option is correct.

Q: Consider the vector space V = R 3 over the field of real numbers F = R. Do the following vectors…

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Q: Let V be an n-dimensional vector space over the field of complex numbers C. Show that V over R has…

A: A complex number z can be written as z=x+iy.Assume a basis for V over C then extend it to V over R.

Q: 1 be given and (ø1, ..., ØN) be the basis of the Let 0 = x1 < ¤2 < ·· space of all continuous…

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Q: 5. Let W be the set of all vectors p(t) in P such that p(0) = 0. Show that W is a subspace of P and…

A: W be the set of all vectors p(t) in P2 such that p(0)=0.