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Today : § 4.2-4.3 : Bases of 6km,
NUKA)
& § 4.5-4.6 : Dimension A Rank
Next : § 4.4 : Coordinates
Homework :
TuesdayMymathlab Homework # 4 : Due TONIGHT by #8pm
My Math Lab Homework # 5 : Due Feb 20 by Hispm
A basis of a subspace H is a collection of vectors
{ 4, Va . .
.
,v.a } < It
that 4) spans It,
and
(2) B linearly independent .
Subspaces are frequently presented as the span of
some collection of vectors ; but those vectors may not
be independent .
How do we find a basis ?
It =
span { / to ) , IG ) , # |If * H
,then
,bosom . xnxx.IR
g- X, yt Xzt + BI
y - t = I= x
, ytxzvt X, ( it - 1)
= span { is.
v. }"nndeewpbspanmyftx,
* 3) is + we B) ± e- span { ⇒set,
i. basis.
Theorem : If H= span { a,
... ,b},
then some collection
of these vectors Is a basis for H.
y A B H I
Eg .
Hsspanlfff:#.
ftp.H.fi#=6kAa=HfaIsfYdmefll:?fI3zj
g , G 93 94 95g b I A ↳{ ↳ b }
a., = -291v. z= -24 four a betq↳ = - g , +293v. 4=-4+24 farCol (A)↳ = 3g , -43* = 3g -2×3
Theorem : The pivotal columns of A form a basis
for Col (A).
( The coefficients expressing
the non - pivotal columns in terms of this
basis can be read off of rref (A).)
' ⇒
HIM→ It.§A → HIM 1 KTEA fma
2 1 bass for Col A).
Caution : The pivotal columns of A,
net rref(A) ,
form a basis for 611A ). Typically
61 1 A) f- 61 ( rref (A) ).
How about Null A) ? How de I find a basis ?
← HE,
Is fedmet to?I?3:)
naff'x¥gff¥IE¥Y¥5fxf #*
ftp.ftxswgfspan.a.bg
lnedly independent.
Theorem : This )procedure for identifying dullA)
always produces a basis for it.
A given subspace always has zillions of bases.
But
there B one thing in common among all bases.
Theorem : Any two bases of a vector space have the
same number of vectors.
Pf. Suppose { a, a } is a bass for a subspace It .
of { g. k,
I }" µ "
So y ,= a
, y tb, yz , g. = azy , tbzyr , D= azyitbzuz
too onsrffr( of;ayof;) §.
the B some nontrivialf¥f=±AI =p
AI = / 91×1 +92×2+93×3biy+bzxtb}x}|=Tx. gtxzttxzv ,
= ×, ( aiui + b
, b) + Xz ( az ↳tKYDT 11319 } bit by .uD
= ( ×, G
, +192 t BAD 91 + ( x,
b,
+ xzbzt Bbdyz = e .
Definition. If V is a vectorspace ,
its dimension
Is the number of vectors in any basis.
Eg.
V= span # ' H' #, ,
We saw that the matrix A =fatty) has 2 pivotal
columns.
V= 61 (A),
so
21
dim ( D= 2
Eg .
B={ polynomials of degrees 3) = { a×3+bx2tcx+d }
Eg .
D= { polynomials }=
span { ×3×?×,
B
= Spas{ 1. x. x3x3,
... ,xH,
... ] DMB -4; dmP= es.