Homework Notes Week 7
Math 24 Spring 2014
6.1#4
(a) Complete the proof in example 5 that , is an inner product (the Frobenius
inner product) on Mnn (F ). In the example properties (a) and (d) have already
been veried, so we need to check that (b) and (c)
Quiz 7
M ATH 24 S PRING 2014
Sample Solutions
Let A =
1+i
1
.
0
1i
1. Compute an eigenvector for the eigenvalue 1 + i.
Solution. The null space of A (1 + i)I =
0
0
1
is spancfw_e1 .
2i
2. Compute an eigenvector for the eigenvalue 1 i.
Solution. The null s
Quiz 6
M ATH 24 S PRING 2014
Sample Solutions
For which triplets of real numbers a, b, c is the matrix
1
0
0
a
1
0
b
c
0
diagonalizable over R? Justify your answer.
Solution. Since the matrix is upper triangular, we see immediately that its characteristic
Quiz 5
M ATH 24 S PRING 2014
Sample Solutions
Let F be an arbitrary eld.
(A) A square matrix A is idempotent if Ak = A for every positive integer k. Show that every idempotent matrix
A Mnn (F ) has determinant 0 or determinant 1.
Solution. Since determina
Worksheet for May 14
M ATH 24 S PRING 2014
Sample Solutions
(A) A the harmonics of a vibrating string of length one are describe by the functions
sin(nx),
n = 1, 2, 3, . . .
on the unit interval [0, 1].
An actual vibrating string will typically overlay se
Slides for May 16
Math 24 Spring 2014
Dual Transformations
Theorem
If T : V W is a linear operator, then T t : W V is a linear
operator too where
T t (f )() = f (T )
for all f W .
Given f , g W , we have
T t (f +g )() = (f +g )(T ) = f (T )+g (T ) = T t (
Homework Notes Week 6
Math 24 Spring 2014
3.4#4b The system
x1 + x2 3x3 + x4 = 2,
x1 + x2 + x3 x4 = 2,
x1 + x2 x3
= 0,
is consistent. To see this we put the matrix
1 1 3
1 2
1 1
2 ,
(A|b) = 1 1
1 1 1
0
0
into reduced row echelon form.
Adding 1 times the r
Homework Notes Week 8
Math 24 Spring 2014
6.4#5 Suppose that T is normal. I will show that T cI is normal too. By
Theorem 6.11(a,b,e), we have
(T cI) = T cI.
Therefore,
(T cI)(T cI) = (T cI)(T cI) = T T cT cT + ccI
and
(T cI) (T cI) = (T cI)(T cI) = T T c