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

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

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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: 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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