4yixt9hyt4i¥z - university of minnesotarylai/docs1/teaching/math... · lecture 22: quick review...
TRANSCRIPT
Lecture 22: Quick review from previous lecture
• If v1, . . . ,vn are nonzero vectors that are mutually orthogonal, meaninghvi,vji = 0 if i 6= j,
mutually orthogonal ) v1, . . . ,vn are linearly independent
• If v1, . . . ,vn is a basis for V and orthogonal (orthonormal), we call they areorthogonal (orthonormal) basis.
[Property:]
• If v1, . . . ,vn is an orthogonal basis in any inner product space V , then for anyvector v 2 V we have
v = a1v1 + · · · + anvn,
where
ai =hv,viikvik2
, i = 1, · · · , n.
—————————————————————————————————Today we will discuss examples for ”Orthogonal(Orthonormal) bases” and then
turn to the Gram-Schmidt process.
- Lecture will be recorded -
—————————————————————————————————Lecture video can be found in Canvas ”Media Gallery”
MATH 4242-Week 10-1 1 Spring 2020
-
-
- -mm
mm YI vill =L,Kien
.
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- -
⇒--1
Example. Consider the orthogonal basis x = (1 1 0)T , y = (1 �1 1)T , andz = (1 �1 � 2)T of R3. Write v = (1 2 3)T as the linear combination of x, yand z.
Example.
(1) The basis 1, x, x2 do NOT form an orthogonal basis.
(2)
p1(x) = 1, p2(x) = x� 1
2, p3(x) = x2 � x +
1
6.
is an orthogonal basis of P (2).
(3) Write p(x) = x2 + x + 1 in terms of the basis p1, p2, p3 in (2).
MATH 4242-Week 9-2 7 Spring 2020
f- 4Yixt9hyt4i¥z .
< r,x > = ( (
'
g ),f ! ) > = I -12=3
11×112 = C ( lol,( j ) ) = 2
.
<v, y > = C ( 'g ) , ft ) ) = I -21-3=2
.
Nyit = L l 't ),l 'T ) ) = 3
.
( V, Z )= - 7
,
112112 = 6.
Thus,
E Ext Ey - Iz#
① for R'"
( co , D )
( I,X > = fo' Ix dx = EM! = % to
,
<x. I > = go'x y' dx = f'd dx = #to
73K :
T✓ 3/23
( co.D )
(2) C P . . P. > = f! I (x - f) dx = IX '- Ext!=O.
( Pi,P, > = fo' l ( x
'-xx# dx = Ix
'- Ix't Ext!
= I - z't I =¥tl= 0 .
[Example Continue]
MATH 4242-Week 10-1 3 Spring 2020
< Pa , Ps) = fo'(x- E) ( x
'- x + f) dx = - -- = 0
.
So, { P , spa
,P, ) is mutually orthogonal,
by Fact,
C Pi,
Pz, P, ) is linearly
independent .Thus, { Pi, Pz
, Pz ) B a basis and
moreover,it is
"orthogonal"
basis. #
(3)- plx) = lpfp.fi?- P , t Pat B- - -
C , Cz CB① C P , P , > = f! 1×7×+1) I dx = Ix't Ix -txt!
= It Etl= f- tf tf
1113112 -_ CP . . P ,> = go', =±M=#
c , = ¥1, = '46. #
③ Cp , Pz > = fjcxtxti ) ( x -E) DX =--
1113112 = fo' c x - hi dx = Eh.
c. = ? -- El
③ CP , Ps ) = folcxztxti ) (x' - Xt 's) dx1113112 = 1/1 80 .
Cs = ① Thus. 12=0113+213+13#
4.2 The Gram-Schmidt Process
Q: How can we construct the orthogonal (or orthonormal) bases?This can be done by the algorithm, known as the Gram-Schmidt process.
§ Given 2 two vectors v,wLet v and w be two vectors, and v 6= 0. How do we make up an vector orthogonalto v so that it forms an orthogonal set that spans the same subspace as span{v,w}?
§ Given 2 orthogonal vectors v1,v2, another vector w
How do we make up an vector v3 orthogonal to both v1,v2? In particular, thevectors {v1,v2,v3} spans the same subspace as span{v1,v2,w}?
MATH 4242-Week 10-1 4 Spring 2020
- -
* (VV,V>= 11411141 cos O .
J = W - LW,D vwa tu .
µ✓ Thus
, J is orthogonal to W.wTIa, Tais -- Cw-T v. D11Wh cos0=11414.fm#it = cw.us - *us= CW = 0
.
Hull J
we call v,the orthogonal projection of
.
Orthogonal projection, of wis sY;u r, taiwan .
--
VE W - f f )Thus
,EV.
,Va, Us ) is orthogonal
span EY , Vz , Vg) = span Erik,w).
§ In general
If v1, . . . ,vn are orthogonal vectors, then for any vector w,
nX
i=1
hw,viikvik2
vi
is the vector nearest to w in span{v1, . . . ,vn}.This is called the orthogonal projection of w onto span{v1, . . . ,vn}.
Furthermore, the vector
w �
nX
i=1
hw,viikvik2
vi
!
is orthogonal to each of v1, . . . ,vn.
§ The Gram-Schmidt process
We start with any basis w1, . . . ,wn for the inner product space V .We then orthogonalize each one to the preceding ones, building up an “orthogonal
basis” as we go.
MATH 4242-Week 10-1 5 Spring 2020
-
Ti
-
O
O-
-
V,= Wi
Va = Wz - cwfgY V,vs = uh - cwnr. -ahem11 Val 12I
un -- Wu-'YYh r. - - - - -9¥Thus
,
IT , -. ., Val is orthogonal ¥
Example. Consider the vectors w1 = (1 1 0)T , w2 = (0 1 1)T , and w3 =(1 0 1)T , that form a basis of R3. To construct an orthogonal basis and anorthonormal basis using the Gram-Schmidt process.
MATH 4242-Week 10-1 6 Spring 2020
- I← -I ii. ¥." ,¥.Yi = wi =L ! ) .isanmh.nu#bas =
Vz = Wz - ( wz , V, > Y
F 114112 = ( ( I ), C ! ) ) = 2
<uh,Vi > =L C ? )
,l ! ) ) =L,
v. = wz - tri -- ( H - it !) --ftUs = W,
-'Y;g ri - 4%9,7-0.Trews
, v.7=441,1 ! ) ) = I,
Cws,us -_ ( ( Kl , s
. ) > = I.
11411'
= 42.
I
a- lil # It I'
= 1¥,)
.
Thus. Y Lal
Example. We know that 1, x, and x2 form a basis for P (2)([0, 1]), the space ofpolynomials of degree 2 on [0, 1]. Let’s turn them into an orthonormal basis,with respect to the usual L2 inner product.
MATH 4242-Week 10-1 8 Spring 2020
-0-
g. =L
9. cxr-x-4.ie?e. f- x - t )Gui -- X
'
- dinging. - 4%9794*2*4Twill continue on Ks j