1. Let V=C[-1,1] with the inner product . Find an orthogonal basis S for the subspace spanned by the polynomials 1, t and t2. The polynomials in this basis S are called Legendre polynomials.
2. Let A be a fixed invertible real nXn matrix. Show that for the vector space Rn, the following formula defines an inner product on Rn. = (Au)T(Av).
(hint: Show the formula satisfy the 4 conditions of the inner product. Note that (Au)T(Av) =(Au)\cdot(Av), and if (Au)\cdot(Au) = 0 ,this means Au= 0 (why?), and it means u= 0 (why?)

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05-12-2022
Mathematics

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1. Let V=C[-1,1] with the inner product . Find an orthogonal basis S for the subspace spanned by the polynomials 1, t and t2. The polynomials in this basis S are called Legendre polynomials.
2. Let A be a fixed invertible real nXn matrix. Show that for the vector space Rn, the following formula defines an inner product on Rn. = (Au)T(Av).
(hint: Show the formula satisfy the 4 conditions of the inner product. Note that (Au)T(Av) =(Au)\cdot(Av), and if (Au)\cdot(Au) = 0 ,this means Au= 0 (why?), and it means u= 0 (why?)

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