MATH 52H HOMEWORK 4 SOLUTIONS
12.13: 5, 6, 7
12.21: 1, 13
Additional Problems:
1. Let F (x, y ) = (u, v ) = (x2 y 2 , 2xy ). Let W be the region dened by 1 x 2 and
0 y 3.
2
(a) What is the relationship between |(x, y )| and |(u, v )|?
(b) Find the area of
MATH 52H HOMEWORK 3 SOLUTIONS
Additional Problems:
1. Let A be an n n real matrix (i.e., the entries are all real numbers). We say that A is
anti-symmetric if At = A. (Here At is the transpose of A.)
(a) Prove that if A is anti-symmetric and if n is odd,
SOME SAMPLE EXAM PROBLEMS
1. Let A be an n n self-adjoint matrix. (That is, assume A = A.) Prove that all the
eigenvalues of A are real.
2. Suppose A is a self-adjoint n n matrix. Suppose u and v are eigenvectors with dierent
eigenvalues. Prove that u and
MATH 52H HOMEWORK 8 SOLUTIONS
1. Suppose V is a 3-dimensional vector space and that T : V V is a linear operator with
det(I A) = 3 52 + 8 4. Suppose that T does not have a basis of eigenvectors.
Find the Jordan form matrix for T .
Solution: Recall the rat
MATH 52H HOMEWORK 9 (Due Friday, March 8)
12
.
1 4
(1) Find a basis of eigenvectors.
x
(2) Find a formula for An
.
y
1. Let A =
Solution: First, det(A rI ) = (1 r)(4 r) (1)2 = r2 5r + 6 = (r 2)(r 3),
so the eigenvalues are 2 and 3. Solving (A 2I )v = 0 gi
MATH 52H MIDTERM SOLUTIONS
February 13, 2002
1(a). Let V be the vector space spanned by the rows of
1
1
A=
2
3
0
1
1
1
2
3
5
7
3
5
8
9
Find a basis for V .
Solution: Put A in rref. Subtract 1,
and 4:
10
0 1
01
01
2, and 3 times row 1 from rows 2, 3,
2
1
1
SOLUTIONS TO THE SAMPLE EXAM PROBLEMS
1. Let A be an n n self-adjoint matrix. (That is, assume A = A.) Prove that all the
eigenvalues of A are real.
Solution: The fundamental property of the adjoint is:
(Au, v) = (u, A v).
Thus if A = A, then
()
(Au, v) =
MATH 52H HOMEWORK 7 SOLUTIONS
1. Let W be the set of vectors in R5 that are perpendicular to (1, 1, 1, 1, 1), (2, 2, 3, 3, 4),
and (3, 3, 4, 4, 5). Find a basis for W .
Solution: Make a matrix A whose rows are the given vectors. Then W is the nullspace
of
MATH 52H HOMEWORK 6 SOLUTIONS
1. Let x0 and x1 be two numbers. Dene xi for i 2 by setting
xn+2 = xn+1 + xn
(for n = 0, 1, 2, . . . ).
(If x0 = 0 and x1 = 1, this gives the Fibonnacci sequence: 0, 1, 1, 2, 3, 5, 8, . . . .)
(a) Find a matrix A such that
xn
MATH 52h HOMEWORK 1 (due Friday, January 11)
1. Suppose we want a way of multiplying two vectors A and B in R2 to get a vector AB ,
and suppose we want the multiplication to satisfy the following laws of algebra:
(1) A(B + C ) = AB + AC and (A + B )C = AC
MATH 52H HOMEWORK 2 SOLUTIONS
1. Let A be an m n matrix. Prove that there is one and only one n m matrix B with
the following property:
(Au) v = u B v
(i)
for every n-vector u and m-vector v. How are the entries of B related to those of A?
Solution 1. Let
MATH 52H HOMEWORK 5 SOLUTIONS
1.13: 1, 12
1.17: 1, 2, 5, 10
Additional Problems:
1. A projection from a vector space V to a subspace W is a linear map T from V to V
such that
i T v W for every v V , and
ii T w = w for every w W .
(a) Prove that if V is ni