Jacobs University Bremen
School of Engineering and Science
Peter Oswald
Fall Term 2010
120201 ESM3A — Problem Set 3
Issued: 22.9.2010
Due: Wednesday 29.9.2010 (in class)
This homework deals with LU factorization, the concept of orthogonality, GramSchmidt orthog
onalization, and QR factorization. Consult the textbook by G. Strang [S] (or any other text on
the subject of matrix calculations and linear algebra) and the script [MK] by K. MallahiKarai.
Do not compute numerical values of square roots, but compute the value of the angle (in radians
or degrees) in Problem 3.2 b).
3.1.
Find the LU decomposition of
A
=

1
2

3
4

5
2

2
4

6
8

1
4

5
6

7
and use it to write down the general solution of the homogeneous underdetermined system
Ax
= 0.
3.2.
Let (
·
,
·
) be the standard inner product on IR
n
.
a) Let
a
= (1
,
2)
T
,
b
= (

1
,
1)
T
∈
R
2
. If
c
is a vector such that (
a
,
c
) =

1 and (
b
,
c
) = 3,
find
c
. b) Find the angle between the vectors
a
= (1
,
2
,
2
,
3)
T
and
b
= (3
,
1
,
5
,
1)
T
. By
definition, the angle between two nonzero vectors
a
,
b
in an Euclidean space
V
with scalar
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 Spring '11
 Prof. Dr. Peter Oswald
 Linear Algebra, Vectors, gramschmidt orthogonalization, Science Peter Oswald, University Bremen School, corresponding orthonormal basis

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