# problemset11 - MAS 213 Linear Algebra II Problem set#11...

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MAS 213: Linear Algebra II. Problem set #11. Tutorial on the 30th of November. This week’s topics: The Gram-Schmidt orthogonalization procedure. Orthogonal projection to a subspace. Application of orthogonal projection to the approximation of continu- ous functions by elementary functions. Tutorial problems: Problem 1: (Problem 6.2.2 from [FIS], various parts.) In the following parts you are given an inner product space V , a set of vectors S V , and a distinguished vector x which is known to lie in the span of S . In each case ﬁnd an orthonormal basis for span( S ), then use the inner product to compute the co-ordinate vector of the given vector x with respect to the basis that you have just determined. Check your answer. (i) V = R 3 with the standard inner product, S = { (1 , 0 , 1) , (0 , 1 , 1) , (1 , 3 , 3) } , and x = (1 , 1 , 2). (ii) V = M 2 × 2 ( R ) with the standard trace inner product, S = {[ 3 5 1 1 ] , [ 1 9 5 1 ] , [ 7 17 2 6 ]} and x = [ 1 27 4 8 ] . 1

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Problem 2: (Problem 6.2.22 from [FIS].) Consider the function f ( t ) = t 2 deﬁned on the interval [0 , 1]. Use an orthog- onal projection to approximate this function by a function of the form ˜ f ( t ) = c 1 t + c 2 t,
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problemset11 - MAS 213 Linear Algebra II Problem set#11...

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