Study Guides (256,120)
CA (124,546)
UTSG (8,506)
MAT (561)
MAT223H1 (102)
N/ A (2)

Summary notes 2End

51 Pages
207 Views

Department
Mathematics
Course Code
MAT223H1
Professor
N/ A

This preview shows pages 1-3. Sign up to view the full 51 pages of the document.
MAT223H1b.doc
Page 1 of 51
Lecture #11 Tuesday, February 10, 2004
ORTHOGONALITY
Definition
Let X, Y be vectors in Rn. The dot product is
R
=
Y
X
Y
X
T.
So, if
=
n
x
x
x
X
2
1
,
=
n
y
y
y
Y
2
1
,
=
=+++=
n
j
jjnn yxyxyxyxYX
1
2211 .
Definition
The length of a vector X, denoted XXX =. So if
=
n
x
x
x
X
2
1
, 0
22
2
2
1+++= n
xxxX.
Example
Let
=
4
2
0
1
X,
=
2
3
2
1
Y. Find
Y
X
and X.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1524322011
=
+
+
+
=
YX.
( ) ( ) ( ) ( )
214201 2222 =+++=X.
Theorem
Let X, Y, Z be vectors in Rn. Then:
1)
X
Y
Y
X
=
. (Proof:
==
=
n
j
jj
n
j
jj xyyx
11
)
2)
(
)
ZXYXZYX
+
=
+
.
3) For a R,
(
)
(
)
(
)
YXaaYXYaX
=
=
.
4) 0X, 00 ==xX.
Example
Let X be a vector in Rn, X 0. Find all Y Rn such that Y is collinear to X and is unitary.
Let
R
=
aaXY,. XaaXY== . We want 1=Y, so X
a
X
aXa11
1±===.
So, X
X
Y1
±= .
www.notesolution.com
MAT223H1b.doc
Page 2 of 51
Definition
Let X, Y be two vectors in Rn. X and Y are orthogonal iff 0
=
YX.
Let
k
XXX ,,, 21 be a set of vectors in Rn.
k
XXX ,,, 21 is orthogonal iff kixi,,2,1,0
=
and jiXX ji = ,0.
Moreover, if kjXj,,2,1,1== , then {X1, X2,, Xk} is orthonormal.
Example
The standard basis of Rn is a orthonormal set.
=
0
0
1
1
E,
=
0
1
0
2
E,
=
1
0
0
3
E.
Definition
If
k
XXX ,,, 21 is an orthogonal set, then
kk XaXaXa,,, 2211 is also an orthogonal set if
kjaj,,2,1,0
=
.
For
j
jX
a1
=, the set
kk XaXaXa,,, 2211 will be orthonormal.
Example
Let
=
1
1
2
1
X,
=
3
3
0
2
X,
=
1
1
1
3
X. Construct an orthonormal set.
321 ,, XXX is an orthogonal set because:
0
i
X.
0
21
=
XX , 0
31
=
XX , 0
32
=
XX .
Find 3,2,1,
=
jaj such that
332211 ,, XaXaXa is orthonormal:
( ) ( ) ( )
6
1
112
11
222
1
1=
++
== X
a,
23
1
2=a,
3
1
3=a.
So,
3
,
23
,
6
3
21 X
XX is orthonormal.
Theorem (Pythagorean)
If X and Y are orthogonal, then 222 YXYX+=+ .
Proof
X and Y are orthogonal; it means 0
=
YX. So,
( ) ( )
222 00 YXYYXXYYXYYXXXYXYXYX+=+++=+++=++=+ .
www.notesolution.com
MAT223H1b.doc
Page 3 of 51
Theorem
Every orthogonal set of vectors of Rn is independent.
Proof
Let
k
XXX ,,, 21 be an orthogonal set. Consider 0
2211
=
+
+
+
=
kk XtXtXtY. We want to prove
that 0
=
j
t.
(
)
00
00
2
22112211
==
=
+
+
+
+
+
=
+
+
+
=
jjj
kkjjjjjkkjj
tXt
XtXXtXXtXXtXtXtXtXYX
.
Theorem
If
n
EEE ,,, 21 is an orthogonal basis of Rn, then n
X
R
,
n
n
nE
E
EX
E
E
EX
E
E
EX
X2
2
2
2
2
1
2
1
1
++
+
=.
Proof
Let X Rn. Then nn EtEtEtX
+
+
+
=
2211 because is a
n
EEE ,,, 21 basis of Rn.
( )
2
2
112211
j
j
jjjjnnjjjjjnnj
E
EX
tEtEEtEEtEEtEEtEtEtEX
==++++=++=.
Example
Let
=
0
1
2
1
X,
=
3
3
0
2
X,
=
1
1
1
3
X. Show that
321 ,, XXX is an orthogonal basis of R3.
It is enough to prove 0
21
=
XX , 0
32
=
XX , 0
31
=
XX because:
321 ,, XXX is an orthogonal set, so X1, X2, X3 are independent.
3dim3=R. So a set of 3 linearly independent vectors is a basis.
Example
Write
=
c
b
a
X as a linear combination of
=
0
1
2
1
X,
=
3
3
0
2
X,
=
1
1
1
3
X.
Since
321 ,, XXX is orthogonal,
3213
2
3
3
2
2
2
2
1
2
1
1
36
3
5
2X
cba
X
cb
X
ba
X
X
XX
X
X
XX
X
X
XX
X+
+
+
+
+
=
+
+
=.
www.notesolution.com

Loved by over 2.2 million students

Over 90% improved by at least one letter grade.

Leah — University of Toronto

OneClass has been such a huge help in my studies at UofT especially since I am a transfer student. OneClass is the study buddy I never had before and definitely gives me the extra push to get from a B to an A!

Leah — University of Toronto
Saarim — University of Michigan

Balancing social life With academics can be difficult, that is why I'm so glad that OneClass is out there where I can find the top notes for all of my classes. Now I can be the all-star student I want to be.

Saarim — University of Michigan
Jenna — University of Wisconsin

As a college student living on a college budget, I love how easy it is to earn gift cards just by submitting my notes.

Jenna — University of Wisconsin
Anne — University of California

OneClass has allowed me to catch up with my most difficult course! #lifesaver

Anne — University of California
Description
MAT223H1b.doc Lecture #11 Tuesday, February 10, 2004 O RTHOGONALITY Definition Let X, Y be vectors in R . The dot product isY = X Y R . x1 y1 x2 y 2 n So, ifX = , Y = , X = x1 1+ x2y 2+ + xny n = xjy j. j=1 x y n n Definition x1 The lengthof a vectorX, denoted X = X X . So ifX = x 2, X = x 2 + x 2++ x 2 0 . 1 2 n x n Example 1 1 0 2 Let X = , Y = . Find X Y and X . 2 3 4 2 X Y = 1)(1 + 0 2) + (2 3( +(4) 2 = 15. X = (1) + 0 ( ) 2 ( + 4)2 = (2) . Theorem n Let X, Y, Z be vectors Rn. Then: n n 1) X Y = Y X . (Proof xjy j= y j j) =1 j=1 2) X (Y +Z )= X Y + X Z . 3) For a R , (aX Y = X aY) = a X Y . 4) X 0 , X = 0 x = 0 . Example Let X be a vector iRn, X 0. Find all R nsuch that Y is collinear to X and is unitary. 1 1 Let Y = aX,aR . Y = aX = a X . We want Y =1 , so a X =1 a = X a = X . 1 So, Y = X . X Page 1 of 51 www.notesolution.com MAT223H1b.doc Definition n Let X, Y be two vectors in R. X and Y are orthogonal iff X Y = 0 . { } n { } Let X 1 X 2 , X k be a set of vectors in . X 1 X 2 , Xk is orthogonaliff xi 0,i =1,2, ,k and X X = 0,i j. i j Moreover, if X j =1, j =1,2,, k , then X 1 X2,, X k is orthonormal. Example 1 0 0 The standard basis ofR n is a orthonormal set.E = , E = , E = . 1 2 3 Definition If {X1, X2,, X k} is an orthogonal s, then {a1X 1a 2 ,2,a X k k } is also an orthogonal seif a 0, j =1,2,,k . j 1 For a j , the set{a1X 1a 2 ,2,a X k k } will be orthonormal. X j Example 2 0 1 Let X 1 = , X 2 = 3 , X3 = 1 . Construct an orthonormal set. 1 3 1 {X1, X2, X 3} is an orthogonal set because: X 0 . i X1X 2 = 0 , X 1 =30, X X =20 3 Find a j,j =1,2,3 such that a 1 1a X2,a2X 3 3} is orthonormal: 1 1 1 1 1 a1= = = , a 2 , a 3 . X1 2 ) + 1 ( ) 1 2 ( ) 6 3 2 3 X X X So, 1 , 2 , 3 is orthonormal. 6 3 2 3 Theorem (Pythagorean) If X and Y are orthogonal, then X +Y 2= X 2 + Y 2 . Proof X and Y are orthogonal; it means X Y = 0 . So, 2 2 2 X +Y = X +Y X +Y = X X + X Y +Y X +Y Y = X X +0+0+Y Y = X + Y . Page 2 of 51 www.notesolution.com MAT223H1b.doc Theorem Every orthogonal set of vectors ofR nis independent. Proof Let {X , X , , X } be an orthogonal set. Consider Y = t X + t X + t X = 0 . We want to prove 1 2 k 1 1 2 2 k k that t j 0 . X jY = 0 X jt X1+t1X +2+t2X k k)= t1X 1 X jt X2 2 ++j X X +j+tjX j k k= 0 . t X 2 = 0 t = 0 j j j Theorem If {E , E , , E } is an orthogonal basis oRn, then X R , n 1 2 n X 1 X E 2 X n X = E 1+ E 2++ En . E 2 E 2 E 2 1 2 n Proof n n Let X R . Then X = t1 1+t E2+2+t E n n because is a {E 1,E2 ,E n} basis ofR . 2 X E j X E j t E1 1 E +2t2E E =nt n E j+t1 1E +j+t E Ej= j E j n n j j j tj= 2 . E j Example 0 1 3 Let X1 = , X 2 = 3, X 3= 1. Show that {X 1,X 2,X 3} is an orthogonal basis of . 0 3 1 It is enough to prove X 1 X 2 0, X 2 X 3 0 , X 1 X 3 0 because: {X ,X , X } is an orthogonal set, sX ,X ,X are independent. 1 2 3 1 2 3 dim R = 3. So a set of 3 linearly independent vectors is a basis. Example 0 1 Write X = b as a linear combination of X = 1 , X = 3 , X = 1 . 1 2 3 3 1 Since {X ,X ,X } is orthogonal, 1 2 3 X X1 X X2 X X3 2a + 3b + a + X = 2 X1+ 2 X 2 2 X 3 = X1 + X 2+ X 3. X1 X 2 X 3 5 6 3 Page 3 of 51 www.notesolution.com
More Less
Unlock Document


Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Unlock Document
You're Reading a Preview

Unlock to view full version

Unlock Document

Log In


OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit