Assignment for Applied Linear Algebra  Math 270
January 28, 2011
The four exercises amount to 15 points. Obtaining 13 points out of 15 guarantees full marks.
The deadline is set to February 14. (Please see details in the Outline of the course, on WebCT.)
Marks.
Just sum the points, no % or letters.
Problem 1
(2 pts.)
Show that
u
1
= (1
,
1
,
1)
T
,
u
2
= (1
,

1
,
0)
T
,
u
3
= (1
,
1
,

2)
T
, form a basis of
R
3
.
Write
v
=
(1
,

2
,
5)
T
in the coordinates of the basis
{
u
1
, u
2
, u
3
}
.
SOLUTION: Three vectors in
R
3
are a basis if and only if they are linearly independent. You
can check this for example by computing
det
1
1
1
1

1
1
1
0

2
= 6
6
= 0
,
or showing that the system
1
1
1
1

1
1
1
0

2
x
1
x
2
x
3
=
0
0
0
has a unique solution, which means that the unique linear combination of
u
1
, u
2
, u
3
is the one
where all coefficients are zero.
(1 pt.)
Solving
1
1
1
1

1
1
1
0

2
x
1
x
2
x
3
=
1

2
5
gives the coordinates of
v
w.r.t. the basis
{
u
i
}
. The coordinates are [4
/
3
,
3
/
2
,

11
/
6]
.
(1 pt.)
Problem 2
(3 pts.)
Recall the definition of linear subspace of a vector space.
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 Winter '09
 Linear Algebra, Algebra, Vector Space, basis, scalar product, AV

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