Drawing planar quadratic, quadratic forms, orthonormal basis and matricies, gram-schmidt orthogonalization process

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Published on 23 Jun 2011
School
UTSG
Department
Mathematics
Course
MAT224H1
Professor
Lecture 9notes byY. Burda
1Drawing aplanar quadric
Weall know what the equation x2+y2=1describes: it is the equation of a
unit circle:
(x, y)
1
It is easy to see what shapeis described bythe equation x2
9+y2
4=1. If
apoint (x, y)lies onthis shape, then the point (x/3,y/2) lies on the unit
circle. So in fact what wehaveis the result of stretching the unit circle bya
factor of 3in the xdirection and byafactor of 2in ydirection. The resulting
shapeis called an ellipse.
(x, y)
(x/3,y/2)
Muchharder is to describehowthe locus of points (x, y)satisfying 3x2+
4xy +3y2=1looks like. In fact wewill spend the next twolectures learning
1
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howto bring this equation to the muchmore manageable form
5x+y
22
+x+y
22
=1
For this equation the coordinates ˜x=x+y
2,˜y=x+y
2seem to beuseful, as
in these coordinates the equationhas the familiar form 5˜x2+˜y2=1, which
describes anellipse.
What is the meaning of these coordinates ˜x
˜y=1/2 1/2
1/2 1/2(x
y)? These
are the coordinates relativeto abasis Bwith [I]B,E=1/2 1/2
1/2 1/2.The vec-
tors from Bare then the columns of [I]E,B=1/2 1/2
1/2 1/21=1/21/2
1/2 1/2.
Thus ˜x, ˜yare the coordinates relativeto the basis v1=1/2
1/2,v2=
1/2
1/2.Now wecan drawour ellipse:
x
y
e1
e2
˜x
˜y
v1
v2
Alternatively wecould havedrawn the ellipse in the plane with coordi-
nates ˜x, ˜yand then the shapeweare after would beits image underthe linear
transformation sending ˜x
˜yto (x
y)=1/21/2
1/2 1/2(x
y). One can verify that this
is the transformation of rotation by45in counterclockwise direction, so we
get the same answer:
2
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˜x
˜y
˜x
˜y
x
y
˜x
˜y! x
y!= 1
21
2
1
2
1
2! ˜x
˜y!
Wewill nowtry to attackthe questions “howto go from the form 3x2+
4xy +3y2to the form 5x+y
22+x+y
22?”, “when drawing in anew system
of coordinates is as easy as drawing in the usual coordinates?” and “howcan
weuse what weknowfrom linearalgebra tohelp us with these questions?”.
2Quadratic forms
The expression 3x2+4xy +3y2is an example of aquadratic form in variables
x, y.Other examples include the forms xy and x2+5y2.
Wecan also form quadratic formsin three variables, for instance x2+
2y2+3z2+5xy +7xz +9yz.
The general definition is the following:
Definition. Areal quadratic form in nvariables is afunction q:RnR
satisfying
q(v)=X
1in
1jn
cijxixj
wherecij aresome fixedreal numbers and x1,...,xnare coordinates of v
relative to the standardbasis of Rn.
Wecan put the coefficients cij from this definition into an n×nmatrix.
It will turn outto bevery convenientto us in the future. For instance we
can rewrite 3x2+4xy +5y2as
x·(3x+4y)+y·(5y)=(xy)3x+4y
5y=(xy)(3 4
0 5) (x
y)
The coefficients in the matrix (3 4
0 5)are in fact the coefficients in the form
3x2+4xy +5y2westarted with: 3is the coefficientat x2,5isthe coefficient
at y2,4is the coefficientat xy.
3
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Document Summary

We all know what the equation x2 + y2 = 1 describes: it is the equation of a unit circle: (x, y) It is easy to see what shape is described by the equation x2. If a point (x, y) lies on this shape, then the point (x/3, y/2) lies on the unit circle. So in fact what we have is the result of stretching the unit circle by a factor of 3 in the x direction and by a factor of 2 in y direction. Much harder is to describe how the locus of points (x, y) satisfying 3x2 + In fact we will spend the next two lectures learning. 1 www. notesolution. com how to bring this equation to the much more manageable form. For this equation the coordinates x = x+y 2. = 1 seem to be useful, as in these coordinates the equation has the familiar form 5 x2 + y2 = 1, which describes an ellipse.