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Solution: APPM 3310 Final Exam Spring 2010
Problem 1. (20 points) Consider the matrix F =
(a) (10 pts) Find the LU-decomposition of F. (b) (5 pts) What is the rank of F? (c) (5 pts) Is F invertible? Why or why not?
Solution:
(a) Here we have U =
(^) and L =
(b) rank(F ) = rank(U ) = 3
(c) det(F ) = det(U ) = − 1 · − 3 · 2 = 6 6 = 0, so F is invertible.
Problem 2. (10 points) Find the least squares solution to the linear system:
x y
Solution:
Note that K = AT^ A =
and f = AT^ b =
and so solving Kx = f yields
x =
Problem 3. (20 points)
(a) (10 pts) Use the Gram-Schmidt process to construct an orthogonal basis of the plane P
spanned by the vectors {w 1 , w 2 } =
(b) (10 pts) Find the orthogonal projection of the vector v =
(^) onto the plane P from
part (a).
Solution:
(a) u 1 =
(^) and u 2 =
(^) − w^2 ·^ u^1 ‖u 1 ‖^2
u 1 =
(b) proj(v) = c 1 u 1 + c 2 u 2 where ci =
v · ui ‖ui‖^2
, so proj(v) =
Problem 4. (20 points) Consider the matrix M =
(a) (4 pts) Find the eigenvalues of the matrix M.
(b) (6 pts) Find the corresponding eigenvectors of the matrix M. (c) (2 pts) Is M complete? Why or why not? (d) (2 pts) Is M positive definite? Why or why not? (e) (2 pts) Find an orthonormal eigenvector basis of R^3 determined by M. (f) (4 pts) M has a spectral factorization M = QΛQ−^1 , write down Q and Λ.
Solution: (a) The characteristic equation here is (2 − λ)(λ^2 − 4 λ + 3) = 0 and so λ 1 , 2 , 3 = 1, 2 , 3.
(b) Here we have v 1 =
, v 2 =
(^) and v 3 =
(c) Yes, since each eigenvalue is distinct and has exactly one eigenvector associated to it.
(d) Yes, since all the eigenvalues are positive definite.
(e) Note that since each eigenvalue is distinct, the corresponding eigenvectors are linearly inde-
pendent and we have w 1 =
√^2 /^2
, w 2 =
(^) and w 3 =
√^0
√^2 /^2
(f) We have Q =
√^2 /^2
(^) and Λ =
Problem 5. (20 points) Consider the linear system
[ A | b ] =
[
]
(a) (5 pts) What is the compatibility condition of the matrix A? (b) (10 pts) Find the general solution of the augmented system given above. (c) (5 pts) Find the solution of minimum Euclidean norm of the system given above.
Solution:
(a) The compatibility condition for any vector b is
[ 1
]
[
b 1 b 2
]
(b) The general solution is x =
2 t − 3 t
for any t ∈ R.
(c) Note that ker(A) =
and we need
2 t − 3 t
= 0 which implies t = 6/5 so the
solution of minimum Euclidean norm is x∗^ =
Problem 6. (20 points) Write “TRUE” or “FALSE”. You do NOT need to justify your answer. Each part is worth 5 points.
(a) A linear transformation L : R^2 → R^2 maps circles to circles. (b) If ‖v + w‖ = ‖v‖ + ‖w‖ then v, w are parallel vectors. (c) Let K be the Gram matrix associated to vectors v 1 , v 2 ,... , vn ∈ V where V is an inner product space, then K is positive definite if and only if v 1 , v 2 ,... , vn span V. (d) The set W = span{x^2 + 1, x^2 − 1 , x^2 + x + 1, x^2 } is a vector space of dimension 4.
