ECES511:
Systems I
Fall 20122013
7 December 2012
1
Dept. of ECE, Drexel University
Assignment 8 Solutions
Written assignment:
1.
Solution
is a linear operator and hence there exists a matrix that is naturally associated with it. We will assume
this representation is w.r.t. the natural basis:
{
}
{
}
.
Thus we can write:
([
])
[
]
[
]
[
]
[
]
…
Eqn [1]
Here, subscript n represents the vector/matrix w.r.t the natural basis.
We want however to find the representation of
with the input and output given w.r.t the basis
.
Note that we know how to get the relationship:
[ ]
[ ]
.
is a matrix such that the
column is
the representation of the basis vectors in
with respect to the basis set
.
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ECES511:
Systems I
Fall 20122013
7 December 2012
2
Dept. of ECE, Drexel University
Thus, representing the first and second column of
w.r.t. the natural basis gives very easily the matrix
as:
[
]
We wish to have the following expression:
([
])
[ ]
,
We can rewrite Eqn [1] above using the fact that
transforms vectors from basis representation in
to
those in
to get:
[
]
[
]
[
]
([
])
If we multiply on the left by
we get:
[
]
([
])
This implies:
[
]
To check if we are correct:
I chose a random vector
.
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 Spring '14
 Linear Algebra, basis, linearly independent columns

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