Prove that any complex inner-product (,)v on a complex vector space V, there is a basis U = {₁,..., un} so that (x, y) v = [y] [x]u In other words for any finite dimensional inner-product s pace, there is a choice of basis, so that with respect to that basis, the inner-product is represented by the standard inner-product.

Q: If {u, v} is a basis for a vector space V, show that {u – 2v, au} is also a basis for V for every…

A: Here we use the definition of basis and linear combination of vectors as follows

Q: isider the complex number 3 a = 4+i√2, where i² = -1. (a) Find the minimal polynomial of a over Q.…

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Q: Let (V, (, )) be an inner product space and let f, g be vectors in V. if || f|| = 4, ||| 5 and || f…

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Q: et V be a vector space and suppose that {v1, …, vn} is a basis for V. Explain why any basis for V…

A: Basis of a vector space :    Let V(F) be a vector space and s be a non empty subset of V.  If  (1)…

Q: Let v = [4 2 -1]" Considering the vector space R³ provided with the inner product defined by: (a, b)…

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Q: 2. Let V be a vector space of dimension n and let B = {v1,..., Un} be a basis of V. (a) Prove that…

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Q: If B is the standard basis of the space P3 of polynomials, then let B = {1,t,t,tº). Use coordinate…

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Q: Consider the set P= {1+2x,-1-x+x*,x+x,1+2x+x}. i. Determine if the set P spans the vector space P, ,…

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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 S = {t+1,t-1,(t-1)2} be a basis of P2, the vector space of all second-degree real…

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Q: (0+0i) 0+0i 1+0i 0+ 0i 0+0i (1+0i is the standard basis for Cn as a vector space over C and again…

A: Basis: Let V be a finite dimensional vector space over a field F. A non-empty subset B of a vector…

Q: its) Let P3:= {ao+ a₁x + a2x² + a3x³|ao, a1, a2, a3 € R} be the vector space of polynomials of…

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Q: Given the operator S on the vector space R₁ [x] given by S (a+bx) = -a+b+ (at 2b) x A) What is SC1)…

A: We know that Pn is the vector space of all the polynomials of degree n+1 and the basis for this…

Q: Let (V, (, )) be an inner product space and let v, w be vectors in V. If (v, w) = 7 and w = 5, then…

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Q: a) Give an example of subspaces U = span(f1, f2, f3) and V = span(g1,92; g3) of RaJa] such that…

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Q: The vectors v' 1 = (4,5,3), v 2 = (8, r, 6) and v’ 3 = (6,7,8) form a basis of R for any real value…

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Q: Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 1, O),…

A: We have to find a basis for the subspace of R³ that is spanned by the given vectors.

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: b) Find the value of k such that (-1,1,1), (1,1,1), (1, -1, k) form a basis of R²

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Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

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Q: = √1 = Let 4] 2 V2 - 2 and 3 = 2 -7 Explain that B = (V1, V2, V3) forms a basis of a 3-dimensional…

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Q: Consider a basis for a space given by vectors (1), (12 + 13x + x²). Note that all vectors must have…

A: consider a basis for  a space given by vectors⊥,12+13x+x2To convert the basis is an orthogonal basis…

Q: The inner product, (alb), of any two vectors a and b in a vector space must satisfy the conditions…

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Q: Show that the set of real polynomials of n – degree (Pn) is a vector space with traditional addition…

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Q: Please discuss your understanding of what a basis is for a vector space

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Q: Check whether the set V= {(2, 3), (3, 4)} is basis of R2 or  not?

A: this a spanning set and linearly independent so the vectors forms a basis of R^2

Q: Let W be the subspace consisting of all vectors 13 a of the form 15 a – 16 b where a and b - are…

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Q: What is the dimension of P₂, the vector space of all polynomials of degree at most two?

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Q: ༤ས་ Let A = = [3 0 0 -3 -2 -1 A basis for the null space of A is { vector

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Q: Consider the basis 8-{(!). (3)} 1 2 B = 2 of R2. What is the coordinate vector of (1) with respect…

A: Given basis is B=12,25 We have to find the coordinate vector of x¯=21 Let the coordinate vector of…

Q: V= घ X, xz LX4 :-X₁ + 2x₂ + −X₂ + 4xy=0 and x₁ + 3x₂ -X₂ - 2xy = 0 Show that V is a subspace of Ru…

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