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: Prove that the vectors (1. 1. 0) (1 2 3) and (2. –1 5) form a basis for R3

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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: Let B= = (1, +2, (x - 1)², 42³) be an ordered basis for P3. Find the coordinate vector of f(x) = 1x³…

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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: (b) The general fact that fcu++eUn]e== c\]j + .. + ca[vn]s allows for rewriting the equation above…

A: It is an important part of statistics. It is widely used.

Q: (a) Prove that the set of all (M) tensors for fixed M, N forms a vector space. (You must define…

A: As per the question, we delve into the world of tensor algebra. In the first part (a), we are asked…

Q: Let V be the set of all ordered pairs of real numbers under the following operations of addition 1.…

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Q: Let V = Poly2 be the vector space of polynomials of degree 2 or less and let W = Poly, be the vector…

A: Suppose we have given a linear transformation ,   is basis of V ,  is basis of W  and associated…

Q: 1) Let v be an element of a vector space V. Prove that -(-v) = v. Your proof should work in any…

A: Here a,b are element of field.

Q: Let the basis {(1,1,0), (0,1,1), (1,0,1)} of the space R^3 be orthonormal in the scalar product…

A: {Here we use  a•(b+c) = a•b +a•c}

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: b) Let Pn be a vector space consisting of polynomials of at most degree n. The standard basis for P,…

A: value of k is 0 see calculation below

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: Explain how fixing a basis v1, corresponding function in n variables of the form q(x1, · ·· , Xn) =…

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Q: (b) Let v₁,..., Um be a list of vectors in a linear space. Define what is understood by saying that…

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Q: (1) Let B = {√₁, 72, 73} be the basis of R³ defined by 1 *-(). *- (). *-(1) ), = = = Find the dual…

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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: Show that the vectors (1, 1, 2), (1,2,5), (5,3,4) of R'(R do not form a basis of R'(R).

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Q: So the  set is not closed under vector addition but is it closed under scalar multiplications and…

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Q: Show that if a vector space V has one basis of n distinct vectors, then every basis for V has n…

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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: If are arbitrary vectors in R2x2, then the mapping A = (A, B) || A|| = ||B| = [a11 a12] a21 a22 A =…

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Q: orthonormal

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Q: Show that B = {v1=(1,0,0), v2=(3,7,-2), v3=(0,4,1)} is a basis for R3

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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: 4. Find representation of the vector v = 2 – 2x+ 6x? with respect to following basis. {1+ 3x + 3x?;…

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Q: ) Consider the vectors -- () --- () --- () a= = Do these vectors form a basis of R³? If yes, prove…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

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: Consider the vector space V over the real numbers, consisting of all polynomials of degree less than…

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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: Show that there do not exist scalars c₁, c₂, and C3 such that c₁(1,0, 1,0) + c2 (1, 0, 2, 1) +…

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Q: Let S = V₁ = {V₁, V2, V3, V4, V5} where, ) 22 = , V3 = ₂V4 = V5 = 22 2 Find a basis for the space…

A: Given: S=v1, v2, v3, v4, v5 v1=13-12, v2=2702, v3=1212, v4=21-20, v5=1221 We have to find a basis…

Q: Is the set of (infinite number of) vectors { (0,0,c) }, where c is a real number, a vector space…

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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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