Practice Final 1 ans

# Practice Final 1 ans - Math 20F Solutions to Final Exam...

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Math 20F Solutions to Final Exam 11:30 AM March 21, 2003 1. Since the matrix is triangular, the eigenvalues are the diagonal entries. Finding eigen- vectors involves ﬁnding bases for the null spaces of A - λI . The answers are λ =1 v = α (1 , 0 , 0) T λ =3 v = β (1 , 1 , 0) T λ =4 v = γ (0 , 0 , 1) T where α, β and γ are nonzero scalars. Any values you chose for those scalars are correct. 2. (a) The easiest way to construct an orthonormal basis is to note that the vectors are orthogonal and each has length 3. This gives us u 1 =( 1 / 3 , 2 / 3 , 0 , 2 / 3 , 0) T u 2 =( 2 / 3 , - 1 / 3 , 0 , 0 , 2 / 3) T u 3 =( 0 , 2 / 3 , 0 , - 2 / 3 , 1 / 3) T . You could use Gram-Schmidt orthogonalization and obtain the same answer. The orthonormal basis is not unique, so you may have obtained a diﬀerent correct answer; however, that’s fairly unlikely since this choice is the obvious one. (b) If the given vector is v , then v T u 1 = 15, v T u 2 = 9 and v T u 3 =3 . Th u s

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Practice Final 1 ans - Math 20F Solutions to Final Exam...

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