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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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 Winter '10
 PeterGibson
 Differential Equations, Linear Algebra, Algebra, Equations, Vectors

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