Biomedical Engineering Reference
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may be contracted horizontally using the summation symbol, thus
r
1
¼
A
1k
t
k
;
r
2
¼
A
2k
t
k
;
...
r
n
¼
A
nk
t
k
:
(A.37)
Introduction of the free index convention condenses this system of equations
vertically,
r
i
¼
A
ik
t
k
:
(A.38)
This result may also be represented in the matrix notation as a combination of n-
tuples,
r
and
t
, and a square matrix
A
,
r
¼
A
t
;
(A.39)
where the dot between
A
and
t
indicates that the summation is with respect to one
index of
A
and one index of
t
,or
2
4
3
5
2
4
3
5
r
1
r
2
:
:
r
n
t
1
t
2
:
:
t
n
2
3
A
11
A
12
:::
A
1n
A
21
A
22
:::
4
5
A
2n
¼
;
(A.40)
:
:
:
:
:
:
A
n1
:
:
:
:
A
nn
if the operation of the matrix
A
upon the column matrix
t
is interpreted as the
operation of the square matrix upon the n-tuple defined by (A.38). This is an
operation very similar to square matrix multiplication. This may be seen easily by
rewriting the n-tuple in (A.40) as the first column of a square matrix whose entries
are all otherwise zero; thus the operation is one of multiplication of one square
matrix by another:
2
4
3
5
r
1
r
2
:
:
r
n
2
4
3
5
2
4
3
5
:
A
11
A
12
:::
A
1n
t
1
:::
0
0
A
21
A
22
:::
A
2n
: : ::: :
A
n1
t
2
:::
0
0
¼
(A.41)
:
:
:
:
:
:
: :::
A
nn
t
n
:
:
:
:
0
The operation of the square matrix
A
on the n-tuple
t
is called a
linear
transformation
of
t
into the n-tuple
r
. The linearity property is reflected in the
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