ENEE 241 02*
READING ASSIGNMENT 5
Tue 02/17 Lecture
Topic:
matrixvector product; matrix of a linear transformation; matrixmatrix product
Textbook References:
sections 2.1, 2.2.1, 2.2.2
Key Points:
•
The matrixvector product
Ax
, where
A
is a
m
×
n
matrix and
x
is a
n
dimensional column
vector, is computed by taking the dot product of each row of
A
with
x
. The result is a
m
dimensional column vector.
•
For a ﬁxed matrix
A
, the product
Ax
is linear in
x
:
A
(
c
1
x
(1)
+
c
2
x
(2)
) =
c
1
Ax
(1)
+
c
2
Ax
(2)
In other words,
A
acts as a linear transformation, or linear system, which maps
n
dimensional
vectors to
m
dimensional ones.
•
Every linear transformation, or linear system,
R
n
→
R
m
has a
m
×
n
matrix
A
associated
with it. Each column of
A
is obtained by applying that transformation to the respective
standard
n
dimensional unit vector.
•
If
A
is
m
×
p
and
B
is
p
×
n
, then the product
AB
is a
m
×
n
matrix whose (
i,j
)
th
element
is the dot product of the
i
th
row of
A
and the
j
th
column of
B
.
Theory and Examples:
1
. A
m
×
n
matrix consists of entries (or elements)
a
ij
, where
i
and
j
are the row and column
indices, respectively. The space of all realvalued
m
×
n
matrices is denoted by
R
m
×
n
.
2
. A column vector is a matrix consisting of one column only; a row vector is a matrix consisting
of one row only. The transpose operator
·
T
converts row vectors to column vectors and vice
versa. By default, a lowercase boldface letter such as
a
corresponds to a column vector. In
situations where the orientation (row or column) of a vector is immaterial, we simply write
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 Spring '08
 staff
 Linear Algebra, Dot Product, linear transformation

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