HINTS TO ASSIGNMENT #4
1.
The linear map
is defined by
2
2
:
R
R
→
T
x
x
T
A
=
)
(
where
A
is a
2
2
×
matrix and
x
is a
column vector. The parametric equation of the line might be written as
.
v
a
x
t
+
=
Applying
T
we get the image
w
b
v
a
v
a
x
T
t
tA
A
t
A
+
=
+
=
+
=
)
(
)
(
which is again the
line. We have used the linearity property. Now, a parallelogram may be written as the
set of points
1
,
0
,
≤
≤
+
+
=
t
s
t
s
w
v
a
x
where
v
and
w
are linearly independent vectors.
By linearity it will be mapped to
w
v
b
x
T
tA
sA
+
+
=
)
(
where vectors
A
v
and
A
w
should be
linearly independent to define a parallelogram. This happens (Linear Algebra!) if and only
if
A
is a nonsingular matrix (i.e. det
A
≠
0).
2.
There are many methods to find a matrix for
T
(vectors onto vectors condition, or line
segments onto line segments condition or vertices onto vertices). Probably the most
straightforward would be to use linear algebra. We know that
T
is determined if we know
the action of
T
on a basis for
2
R
and vectors (1,0) and
(0,1) constitute such a basis. We
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 Fall '09
 RomauldStanczak
 Calculus, Linear Algebra, Vector Space, lim, inverse image

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