MATH 423500/200
March 30, 2012
Test 2: Solutions
Problem 1 (20 pts.) Find the determinant
011
1 0 1
A = 1 1 0
1 1 1
111
of the matrix
11
1 1
1 1 .
0 1
10
Solution: det A = 4.
Let us modify the rst row of A adding to it all other rows. These elementary row
MATH 423200/500
Spring 2012
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) = a + bx + cx2 . The
MATH 423500/200
May 7, 2012
Final exam (with solutions)
Problem 1 (15 pts.)
and p(3) = 2p(2).
Find a quadratic polynomial p(x) such that p(1) = 2, p(2) = 5,
Solution: p(x) = x2 + 1.
Problem 2 (20 pts.) Let V and W be subspaces of the vector space Rn such
MATH 423200/500
April 13, 2012
Quiz 1: Solution
Problem. Let V be a subspace of R4 spanned by the vectors x1 = (1, 2, 2, 0), x2 = (1, 2, 3, 2),
and x3 = (1, 0, 5, 1).
(i) Find an orthogonal basis for V .
Let us apply the Gram-Schmidt orthogonalization pro
MATH 423200/500
April 20, 2012
Quiz 2: Solutions
Problem 1. Let A be a square matrix with real entries. Suppose that A is both skew-symmetric
and orthogonal. Show that A has no eigenvalues other than i and i. Section 200 students, also
show that i and i a
MATH 423200/500
April 27, 2012
Quiz 3: Solution
Problem. Let R denote a linear operator on R3 that acts on vectors from the standard basis
as follows: R(e1 ) = e3 , R(e2 ) = e1 , R(e3 ) = e2 .
(i) Is R a rotation about an axis? Is R a reection in a plane?
MATH 423-200/500
Spring 2012
Sample problems for Test 1
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.) Let P3 be the vector space of all polynomials (with real coecients)
of degree at most 3. Determine which of the following
MATH 423-200/500
Spring 2012
Sample problems for Test 1: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.) Let P3 be the vector space of all polynomials (with real coecients)
of degree at most 3. Determine which of the
MATH 423-500/200
February 17, 2012
Test 1: Solutions
Problem 1 (20 pts.) Determine which of the following subsets of the vector space R3
are subspaces. Briey explain.
(i) The set S1 of vectors (x, y, z ) R3 such that xyz = 0.
(ii) The set S2 of vectors (x
MATH 423200/500
Spring 2012
Sample problems for Test 2
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.)
p(1) = 0, and p(2) = 4.
Find a cubic polynomial p(x) such that p(2) = 0, p(1) = 4,
Problem 2 (25 pts.) Evaluate a determina
MATH 423200/500
Spring 2012
Sample problems for Test 2: Solutions
Any problem may be altered or replaced by a dierent one!
Problem 1 (20 pts.)
p(1) = 0, and p(2) = 4.
Find a cubic polynomial p(x) such that p(2) = 0, p(1) = 4,
Let p(x) = a + bx + cx2 + dx3
MATH 423200/500
Spring 2012
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.) Consider a linear tr