Today : § 1.4 : The Matrix Equation Aek

& § 1.5 : Solution Sets

Next : § 1.7 : Linear Independence

Reminders :

My Math Lab Homework # 1 of # 2 : Due Men,

Jan 22

MATLAB Homework # 1 : Due TONIGHT 1

Theorem : Let A be an mxw matrix.

The following four statements are equivalent.

(a) A x. = b. can be solved for I,

for any b. e Rm .

(b) Every vector b. EIRM is a linear combination

of the columns of A.

(c) The columns of A span Rm.

(d) A has a pivot in each row .

Eg .

Let A =

#I

,If } ) .

Do the columns of A spanR3 ?

rew&P§ rref ( MATLAB )

- 4/3 -413

A =L's! ;io1Question : Is /§/ in the span of the columns of A ?KEYIiftp.E.IM#*li3*ili9

1.5 Solution Sets of Linear Systems

Def .

A system of equations is homogeneous ifit has the form A ± =p zero vector ( ¥ )

3×+5*-4×3=0⇒ 1.3.52µ §gfl8)

. 3×, -2×2+4×3 = 0

¢ system of equations matrix equation §s u

xpttxtsfxsffffd ⇒ 13144181

vector equation augmented matrix

Key observation : there is always at least one solution : x. =p.

But there may be mere.

I :{tiled → 635daL → toiii. → patted → Helbigrename

→ § free×

' ii.If µaftmg*fHosts Ml →spandaulisted8k¥49'

Iiidtiiinsxol ¥I¥t¥¥f

parametric solution

smutty =ttµFELl¥ffI¥y

Inhomogeneous systems are linear systems that are net homogeneous.

it.

A x. =b where b. to.

egl :

;taigl→l'ftp.l-teiiiiiil

s :b"

f.l¥*dt÷¥.tt?t.#.ytH+HnwtF.YIri

-

compare to \ \? #

Mfg;D,

I

:ihi:lnl¥*fsH + TH + a

'⇒ HEIM"yd=td ; lY¥

wesawks.yi.gl#ldgIobiiodE..fMgsl↳ s

Wealsesaw =s(4Y|

KEY !

.tl#ldfIobiiflNosdutrn!

Theorem .Let A be a matrix

.

Denote the (parametric)solution set of the homogeneous equation

A ± = e

as Vh.

(a) Even though A x. = e is always consistent ( I =p

is always in vn ),

for given b. ¥ e , the equation

A x. = b. may be Inconsistent.

(b) If A x. = b. is consistent lie.

b. is in the span of the

columns of A ) and if x. =p is any particularsolution

,then the general solution is x. =p + vh .