Review and Examples
(Spr. 2006)
Due Thursday 6th April
These are examples of questions with the degree of difficulty that you will
encounter in the labexam. Make sure that you can do them!
Use Mathematica to solve all parts of the problems
Vectors
Problem 1.
Show that for any vectors
~a
,
~
b
,
~
c
that,
(i)
~a.
(
~a
∧
~
b
) = 0
(ii)
~a.
(
~
b
∧
~
c
) = (
~a
∧
~
b
)
.~
c
Lists and Matrices
Problem 2.
Display the following matrix in matrixform.
matrix
=
{{
0
.
2
,

0
.
4
,
0
.
1
}
,
{
0
.
4
,
0
.
2
,

0
.
3
}
,
{
0
.
6
,

0
.
3
,

0
.
1
}}
We haven’t found the eigenvalues and vectors of symmetric matrices be
fore; however, this illustrates the power of mathematica. Look up Eigenvalue
in the help manual. You won’t fully appreciate the power of this until you
have to solve eigenvalue problems by hand in advanced mechanics and quan
tum mechanics.
Find the determinant, eigenvalues and eigenvectors of the matrix
matrix
.
Show that the sum of the eigenvalues (= the trace of the matrix) and the
product of the eigenvalues (= the product of the eigenvalues) are both real
even though the eigenvalues themselves are complex.
Function definitions
f
[
x
]
Problem 3.
Define the function
f
(
x
) = 0
.
8
x
+
x
2

3
.
2
x
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 Spring '08
 DUXBURY
 Physics, Linear Algebra, Matrices, Work, Eigenvalue, eigenvector and eigenspace, Fundamental physics concepts, Orthogonal matrix

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