LinAlgExercises - SC/MATH 2271 3.0 Differential Equations...

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Unformatted text preview: SC/MATH 2271 3.0 Differential Equations for Scientists and Engineers Winter 2010 Practice problems in linear algebra 1. Consider the following set of vectors in R3 : √ 1 2 √ √ S = 2 , 1 + 2 , √ 1 2 1 2 . 1 (a) Are the vectors linearly independent, or linearly dependent? Justify your answer. (b) Letting W denote the span of S , show that W is a vector space. (c) Determine the dimension of W . (d) Find two different orthonormal bases of W . (e) Determine the closest point y in W to the vector 1 1 . x= 0 (f) What is the distance from x to W ? (Hint: compute ||x − y ||.) 2. Consider the vector space V = the set of all continuous functions f : [0, 1] → R. Let f, g, h ∈ V be defined by the formulas f (x) = 2x, g (x) = x2 , and h(x) = 1. (Note that the zero vector in V is the zero function, z (x) = 0.) (a) Show that f and g are linearly independent. (b) Find the angle between f and g . (c) Let W denote the span of {f, g, h}. What is the dimension of W ? (d) Find the closest point y in W to the vector cos x ∈ V . (Note that y may be viewed as the best possible quadratic approximation to cos x over the interval [0, 1].) 1 ...
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