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Math 337 – Fall 2004 Final Examination Preparation
Instructions. These problems are provided to assist you to prepare for the final exami- nation. These problems are intended to supplement your review of the homework, lectures, reading and midterm examinations. Problems marked with an asterisk () are among the more difficult that might be expected on the final. Those with two asterisks (*) are espe- cially challenging.
Problem 1. Suppose
A =
[
]
B =
(^) x =
[
]
u =
[
]
v =
Compute each expression below if possible:
(a) AB (b) BA (c) A^2 + B^2 (d) At^ (e) BtB (f ) BBt
(g) Ax (h)
v B
(i) xtx (j) xxt^ (k) xtAx (`) x · u (m) u · v
(n) ||x||^2 + ||u||^2 − ||v||^2 (o)
x v
(p) 3x − 2 u (q) 2v + A (r) vtB
Problem 2. Suppose A is a 3 × 5 matrix. For each part below, give the matrix B such that BA is the matrix C described.
(a) C is obtained from A by adding 3 times the first row to the third row. (b) C is obtained from A by subtracting the third row from the first row. (c) C is the 2 × 5 matrix having only zeros as components. (d) C is obtained from A by multiplying its second row by 7. (e) C is obtained from A by exchanging its second and third rows.
Problem 3. Find the LU factorization of the following matrices:
(a)
[
]
(b)
[
a b b a
]
(c)
(^) (d)
(e)
(f )
a 0 0 0 2 a 0 0 2 2 a 0 2 2 2 a
Problem 4. Find the general solution of the following systems of equations:
(a)
x y z
(^) (b)
[
] [
x y
]
[
]
(c)
x y z w
(d)
x y z u v
Problem 5. For each of the following matrices, find its inverse matrix if possible.
(a)
[
]
(b)
[
cos x sin x − sin x cos x
]
(c)
(^) (d)
a 0 −a a^2 0 a^2
Problem 6. For each of the following matrices, find a basis for the four fundamental subspaces. Also, find a basis for the fundamental subspaces for each matrix in problem 2.
(a)
[
]
(b)
[
]
(c)
[
]
(d)
[
]
Problem 7∗. Determine if the set of vectors S is a basis for the null space of the matrix A where
S =
A =
Would S be a basis if it contained additionally the vector (1, 0 , 0 , 0)? Why or why not?
Problem 8∗∗. Consider the two-dimensional subspace V of R^3 having the orthonormal basis {a 1 , a 2 }. Let P be the matrix that implements projection into V ; that is, P x is the point in V closest to x. Suppose u and w are independent vectors in R^3 with P u and P w dependent. Show there are scalar values α and β such that αu + βw 6 = 0 and αu + βw is in the left null space of the matrix having a 1 and a 2 as columns.
Problem 17. For each matrix below find all eigenvalues and as many independent eigen- vectors as possible. Diagonalize the matrix using an orthogonal matrix when the matrix is symmetric.
(a)
[
]
(b)
[
a b b a
]
(c)
(^) (d)
(e)
[
]
(f )
[
a a^2 a^2 a^3
]
(g)
(^) [ 1 3 −1 ] (h)
1 0 a 0 2 0 a 0 1
Problem 18∗. Suppose that {q 1 , q 2 , q 3 , q 4 , q 5 } is an orthonormal basis (of column vec- tors) for R^5. If possible, find a basis of eigenvectors for R^5 along with the corresponding eigenvalues for the matrix A given by
A = 3 q 1 qt 1 + 2q 2 qt 2 + 4q 3 qt 3.
If possible, diagonalize A.
Problem 19∗. Consider the sequence x 0 = 0, x 1 = 1, x 2 = 32 , x 3 = 74 , x 4 = 158... which satisfies the relation
xn+2 = =
xn+1 −
xn.
(a) Find the matrix A such that [ xn+ xn+
]
= A
[
xn+ xn
]
(b) Give a formula for (xn+1, xn) in terms of powers of A and (x 1 , x 0 ) = (1, 0). (c) Find the eigenvalues and eigenvectors of A. (d) Diagonalize A. (e) Find a explicit formula for xn. (f) Which of the following values best approximates x 314 : 1, 2, π, 432 or 8. 315 · 1012.
Problem 20. Which of the following transformations of R^2 are linear?
(a) T (v) = (v 1 , v 2 ) (b) T (v) = (|v 1 |, 0) (c) T (v) = (3v 1 , v 2 − 3 v 1 ) (d) T (v) = (0, 0)
Problem 21. If possible find matrices satisfying the description.
(a) The matrix transforms (1, 0) into (2, 4) and transforms (0, 1) to (1, 2). (b) The matrix transforms (2, 4) to (1, 0) and (1, 2) to (0, 1). (c) The matrix transforms (2, 4) to (2, 0) and (1, 2) to (1, 0).
Partial Solutions
Note well: These solutions are not complete. There are intended only to indicate whether you have the right approach and solution. On the examination you will be expected to provide complete solutions to problems to receive credit.
Solution 1.
(a) undefined (b)
2 1 0 0 1 2
(^) (c) undefined (d)
[ 2 1 1 2
] (e)
[ 1 0 0 1
] (f )
1 0 0 0 0 0 0 0 1
(g)
[ 5 4
] (h) undefined (i) [5] (j)
[ 4 2 2 1
] (k) [14] (`) 0 (m) undefined
(n) − 4 (o) undefined (p)
[ 4 7
] (q) undefined (r) [ 1 3
]
Solution 2.
(a)
1 0 0 0 1 0 3 0 1
(^) (b)
1 0 − 1 0 1 0 0 0 1
(^) (c)
[ 0 0 0 0 0 0
] (d)
1 0 0 0 7 0 0 0 1
(^) (e)
1 0 0 0 0 1 0 1 0
Solution 3.
(a)
[
] [
]
(b)
[
b a 1
] [
a b 0 a − b
2 a
]
(c)
1 2 1 0 0 23 1
(d)
1 0 0 4 1 0 7 2 1
1 2 3 0 − 3 − 6 0 0 1
(^) (e)
1 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 23 1 0 0 0 0 0 0 34 1 0 0 0 0 0 0 45 1 0 0 0 0 0 0 56 1 0 0 0 0 0 0 67 1
2 1 0 0 0 0 0 0 32 1 0 0 0 0 0 0 43 1 0 0 0 0 0 0 54 1 0 0 0 0 0 0 65 1 0 0 0 0 0 0 65 1 (^0 0 0 0 0 0 )
(f )
2 a 1 0 0 2 a
2 2 a^1 a
2 a
2 a 1
a 0 0 0 0 a 0 0 0 0 a 0 0 0 0 a
Solution 4.
(a) no solution (b)
[
]
[
]
Solution 10. (z · x)x + (z · y)y
Solution 11. b = t + / 3
Solution 12. a = 1, b = − 1
Solution 13.
P =
1 2 0
1 2 0 1 0 1 2 0
1 2
(^) λ 1 = 1, λ 2 = 1, λ 3 = 0 v 1 =
(^) , v 2 =
(^) , v 3 =
Solution 14. A basis for the null space of A is {(− 1 , 1 , 0 , 0), (− 1 , 0 , 1 , 0), (− 1 , 0 , 0 , 1)}. Gram-Schmidt gives (^)
√^2 /^2
√ 6 6 −
√ 6 √^6 6 3 0
√ 12 12 −
√ 12 12 −
√ 12 √^12 12 4
Solution 15. (^) [ (^) √ 2 2
√ 2 2 −
√ 2 2
√ 2 2
] [ √
]
Solution 16.
(a) − 1 (b) 24 (c) 0 (d) 1 (e) − 2 a^3 (f ) 1 (g) 6 (h) 0
Solution 17. Note: Answers are not unique.
(a)
[
i −i 1 1
] [
i 0 0 −i
] [
− i 2 12 i 2
1 2
]
(b)
[ √
2 2 −^
√ 2 √^2 2 2
√ 2 2
] [
a + b 0 0 a − b
] [^ √ 2
2
√ 2 2 −
√ 2 2
√ 2 2
]
(c)
√ 2 2 0 −
√ 2 2 √^0 1 2 2 0
√ 2 2
1 0 0 0 1 0 0 0 − 1
√ 2 2 0
√ 2 2 0 1 0 −
√ 2 2 0
√ 2 2
(d)
1 1 1 0 1 1 0 0 1
1 0 0 0 2 0 0 0 3
1 − 1 0 0 1 − 1 0 0 1
(e) 2,
[ 1 0
] (f )
[ ( a^2 + 1
)− 1 / 2 a
( a^2 + 1
)− 1 / 2
a
( a^2 + 1
)− 1 / 2 −
( a^2 + 1
)− 1 / 2
] [ a + a^3 0 0
] [ ( a^2 + 1
)− 1 / 2 a
( a^2 + 1
)− 1 / 2
a
( a^2 + 1
)− 1 / 2 −
( a^2 + 1
)− 1 / 2
]
(g)
(h)
√^0 1
1 + a 0 0 0 2 0 0 0 1 − a
√^0 1
Solution 18.
A = [ q 1 q 2 q 3 q 4 q 5 ]
[ q 1 q 2 q 3 q 4 q 5 ]T
Solution 19.
(a) A =
[
]
(b)
[
xn+ xn
]
= An
[
x 1 x 0
]
(d)
[
] [
] [
]
(e) 2 − 21 −n^ (f ) 2
Solution 20. (a) linear, (b) not linear, (c) linear, (d) linear
Solution 21.
(a)
[
]
(b) no such matrix (c)
[
]
[
2 α −α 2 β −β
]
