MATH 304502/506
Fall 2011
Sample problems for Test 2: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.) Let M2,2 (R) denote the vector space of 2 2 matrices with real
entries. Consider a linear operator L : M2,2 (R) M2
MATH 304502/502/506: PROBLEM SET 6
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 4.1.4
Let L : R2 R2 be a linear operator. If
1
2
L
=
,
2
3
1
L
1
5
=
,
2
find the valu
MATH 304502/502/506: PROBLEM SET 5
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 3.5.5
be the standard ordered basis of R3 and let = u1 , u2 , u3 for
Let = cfw_i, j, k
MATH 304502/502/506: PROBLEM SET 9
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 5.5.32
Find the best least squares approximation to f (x) = x on [, ] by a trigonometric
MATH 304502/502/506: PROBLEM SET 7
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 5.1.10
Find the distance from the point (1, 1, 1) to the plane 2x + 2y + z = 0.
Problem 2:
MATH 304502/502/506: FINAL REVIEW PROBLEMS
Problem 1
Consider the matrix
4 2
5 2
1 9
2
A= 1
4
(a) Find the LU decomposition of A.
(b) Use (a) to find det(A).
Hint: If P and Q are n n matrices, what is the relation between det(P Q) and det(P ) det(Q)?
(c)
MATH 304502/502/506: PROBLEM SET 2
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 1.4.7
Let
1
2
21
A=
21
1
2
.
Compute A2 and A3 . What will An turn out to be?
Problem 2:
MATH 304502/502/506: PROBLEM SET 8
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 5.4.2
T
Let x = (1, 1, 1, 1) and y = (8, 2, 2, 0)T .
(a) Determine the angle between x and
MATH 304502/502/506: PROBLEM SET 3
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 3.2.1.(b)
Determine whether the set
x1
x2
R x1 x2 = 0
2
forms a subspace of R2 .
Problem
MATH 304502/502/506: PROBLEM SET 3
Each problem is worth 10 points, and five of the following problems will be graded for a total of 50 points.
Problem 1: Exercise 3.3.2.(c)
Determine whether the vectors
2
1 ,
2
3
2 ,
2
2
2 ,
0
are linearly independe
MATH 304502/506
Fall 2011
Sample problems for the nal exam
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.) Find a quadratic polynomial p(x) such that p(1) = p(3) = 6 and
p (2) = p(1).
Problem 2 (20 pts.) Let v1 = (1, 1, 1), v2
MATH 304502/506
Fall 2011
Sample problems for the nal exam: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.) Find a quadratic polynomial p(x) such that p(1) = p(3) = 6 and
p (2) = p(1).
Let p(x) = ax2 + bx + c. Then p
MATH 304502/506
Fall 2011
Sample problems for Test 1
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.)
and p(3) = 7.
Find a quadratic polynomial p(x) such that p(1) = 1, p(2) = 3,
1 2
4
2
3
2
Problem 2 (25 pts.) Let A =
2
0 1
2
MATH 304502/506
Fall 2011
Sample problems for Test 1: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.)
and p(3) = 7.
Find a quadratic polynomial p(x) such that p(1) = 1, p(2) = 3,
Let p(x) = ax2 + bx + c. Then p(1) =
MATH 304502/506
Fall 2011
Sample problems for Test 2
Any problem may be altered or replaced by a dierent one!
Problem 1 (15 pts.) Let M2,2 (R) denote the vector space of 2 2 matrices with real
entries. Consider a linear operator L : M2,2 (R) M2,2 (R) give
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