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Unformatted text preview: √
1/√2
−1/ 2
√
b1 =
, b2 =
.
1/ 2
1/ 2
The proof of the theorem of Section 2.1 shows that these vectors are orthogonal
to each other, and hence the matrix
√
√
1/√2 −1/ 2
√
B=
1/ 2 1/ 2
is an orthogonal matrix such that
2
0 B T AB = 0
8 . Note that B represents a counterclockwise rotation through 45 degrees. If
we deﬁne new coordinates (y1 , y2 ) by
x1
x2 =B y1
y2 , equation (2.6) will simplify to
y1 y2 or
2
2
2y1 + 8y2 = 2
0 0
8 y1
y2 = 1, 2
2
y1
y2
√
√
+
= 1.
2
(1/ 2)
(1/ 8)2 We recognize that this is the equation of √ ellipse. The lengths of the semimajor
an
√
and semiminor axes are 1/ 2 and 1/(2 2). 38 0.4
0.2 0.4 0.2 0.2 0.4 0.2
0.4 Figure 2.1: Sketch of the conic section 5x2 − 6x1 x2 + 5x2 − 1 = 0.
1
2
The same techniques can be used to sketch quadric surfaces in R 3 , surfaces
deﬁned by an equation of the form a11 a12 a13
x1
x1 x2 x3 a21 a22 a23 x2 = 1,
a31 a32 a33
x3
where the aij ’s are constants. a11
A = a21
a31 If we let
a12
a22
a32 a13
a23 ,
a33 x1
x = x2 ,
x3 we can write this in matrix form
xT Ax = 1. (2.7) According to the Spect...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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