Graphics Programs Reference
In-Depth Information
Solution
1
2
∂
f
1
x
2
2
x
x
2
x
∂
f
y
x
=
=
y
=
∂
x
2
+
y
2
∂
x
2
+
y
2
+
y
2
+
y
2
x
y
2
)
T
(
x
2
y
2
)
(
x
2
∇
f
(
x
,
y
)
=
/
+
y
/
+
2
T
∇
−
,
=
f
(
2
1)
−
0
.
40
.
2
f
(
x
2
y
2
)
x
2
y
2
∂
+
−
x
(2
x
)
−
+
=
=
∂
x
2
(
x
2
+
y
2
)
2
(
x
2
+
y
2
)
2
∂
2
f
x
2
−
y
2
=
∂
y
2
(
x
2
+
y
2
)
2
2
f
2
f
∂
∂
−
2
xy
=
x
=
∂
x
∂
y
∂
y
∂
(
x
2
+
y
2
)
2
−
x
2
+
y
2
−
2
xy
1
H
(
x
,
y
)
=
x
2
y
2
(
x
2
y
2
)
2
−
2
xy
−
+
−
0
.
12 0
.
16
H
(
−
2
,
1)
=
0
.
16 0
.
12
A2
Matrix Algebra
Amatrix is arectangular array of numbers. The
size
of amatrix is determinedbythe
number of rows and columns, also called the
dimensions
of the matrix. Thus amatrix
of
m
rows and
n
columns issaid to have the size
m
n
(the number of rows is always
listedfirst). A particularly important matrix is the square matrix, which has the same
number of rows and columns.
An array of numbers arrangedinasingle column iscalleda
column vector
,or
simplyavector. If the numbers are set out in a row, the term
row vector
is used. Thus
a column vectoris amatrix of dimensions
n
×
×
1 and a rowvector can be viewedas a
matrix of dimensions 1
n
.
We denote matrices by boldface, upper case letters.Forvectors we use boldface,
lower case letters. Here areexamples of the notation:
×
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
b
1
b
2
b
3
A
=
b
=
(A9)
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