E.g: In}Lecture 16, § 2.8 contd. Recall a subset H of R" is a subspace if-F in H.-I,J' in H 㱺...
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Lecture 16, § 2.8 contd. Recall a subset H of R" is a subspace if - F in H. - I, J' in H ⇒ Ity' in H. - I' in H and c a scalar ⇒ CI in H. E.g: Span {Ti,,...> In} is a subspace. Column space: The dpan of the columns of m x n matrix A is column space CDA of A. Null space : The solutions of homogeneous system A-I = F is a subspace of 112" (if A has size mxn) called null space NWA of A. Linear independence: A set of vectors {Ii,,..., Tin} is linearly independent if C, Ii, t... + cretin = 0 ⇒ c, = . . . = en-0 if and only if A I = 8 has ONLY trivial dolution where A = Iii, . . . In] if and only if echelon form of A (not the augmented matrix [A 181) has a pivot in every column. Today, Basis of a subspace o. Let It be a subspace of Them. A collection of vectors Ti,,..., Tin in H is a basis of H if ① Span {it,...,Tin} = H. ② {Ti,..., Tin} is linearly independent.
Transcript of E.g: In}Lecture 16, § 2.8 contd. Recall a subset H of R" is a subspace if-F in H.-I,J' in H 㱺...
Lecture 16, § 2 . 8 contd.
Recal l a subset H of R " i s a subspace i f- F i n H .
- I , J' i n H ⇒ I t y ' i n H .
- I ' i n H and c a scalar ⇒ C I i n H .
E.g : Span{Ti,,...>I n } i s a subspace.
Column space : The dpan of the columns of m x n matrix A i s column spaceCDA of A .
N u l l space : The solutions of homogeneous system A-I = F i s a subspace
of 112" (if A has size mxn) ca l led n u l l space N WA of A .
Linear independence: A set of vectors {I i , , . . . , Tin} i s linearly independent i fC, I i , t . . . + cretin = 0 ⇒ c , = . . . = e n - 0
i f and only ifA I = 8 has ON LY t r i v ia l dolution where
A = I i i , . . . I n ]
if and only i fechelon form of A (not the augmented matrix [ A 181)has a pivot i n every column.
Today,
Ba s i s of a subspace o. Let I t be a subspace of Them. A collection of vectorsTi,,..., Tin i n H i s a basis of H i f
① Span { i t , . . . , Tin} = H .
② {T i , . . . ,Tin} i s linearly independent.
E.g : (Special basis of Rn) Consider R " a s a subspace of itself.Then R " has a special basis consisting of
t h e r e d ' I i = Yo;) fine..it;¥a.. for " " ' " '
" '
I n d e e d , any 1¥;)
i n R n c a n be written.
gpa.fi#..t..?eEnzt=fhnt.'
" i n .
Also, t h e matrix A - [E; E . . . En?
=/!!..-÷) = I n
has a pivot i n every column.
Theorem: Every dubspace H of 112" has a basis, though not unique. Moreover
any 2 bases of H have t h e same#of vectors. T h i s # i s called the
dimension of H .
Goal : F i n d basis of Cola a n d Nut A .
Basis of Col A : ① Express A i n echelon form (REF n o t needed).
② The pivot columns of A form a basis of ColA .11(NOT O F E .F . O FA)
E x e r : F i n d a basis for ColA where A = (§ §%,}g-Ig)
-
Solution: I 4 8 - 3 - 7 Ra t>R a TR ,t:::::.li#::il::::÷÷÷:LI, R a → 122/3
I:::÷÷÷.li#ii::::l:::::.:.i:...l/yl2uHRut4R3{§ §§, §. I;)
% Bas is Y ' A = {pivot columns of A}
t.i.H.tt:Dooo Cola i s a subspace of 1124 of dimension 3 .
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Basis of N u l A : ① Solve A I = 8 .
② Express general solution i n parametric form.③ T h e vectors associated with parameters give a basis of Nula.
E x e r : F i n d N u l A of the matrix A from previous exerc ise.
Solution : Need t o solve A I = D . We s aw t h a t the E F of A i s
l:÷÷÷÷,§,Linear system
X , t 4×2 t 8×3 -3×4-7×5 = 0
2×2 t 5×3 - X s = O
- Xu -4×5 = 0 .
B A S I C : X i , X i , X 4 ; F ree : X s , X s .
× , = -4×5 .2×2 = -5×3 t X s ⇒ × 2 = - { x , t {Xs-
X , = -4×2-8×3+3×4-17×5= -41-Ex, t {xs) -8×3 t3f4xs) t 7× 5= 10×3-2×5 -8×3-12×5+7×5 = 2×3 - 7 × 5
To General lolution =2×3-7×5 Parametric 2×3 - 7 × 5
{" "I} + I} ¥ ("I}) t(0×5)- 4×5 O -4×5
X s O X s
= X 3 2 t X s - 7
"'t.it:÷ii÷:p:i÷:p.
% dimension of Nu l A = 2 .
Note from previous exercises, dim. of colA t dim. Nu l A = 3 + 2
= 5
= #of columns ofA .
This i s always t h e case :
thank-Nullity : I f A i s m x n matrix, then dim. colA t d im N u l a = n
=#of columnsof A .
Will learn m o r e about th i s next t i m e . My OH for Mon, b e d a r e cancelled.
