Range and Null Space

Range and Null Space

Stephen Boyd and Sanjay Lall EE263

Stanford University

1

Nullspace of a matrix

I

the nullspace of

is defined as

litres KitXzes es aeriaxies

null

null is set of vectors mapped to zero by null is set of vectors orthogonal to all rows of

I

AI A xuxa A

O

AKA LAX O

E

null gives ambiguity in given

:

if

and null , then

conversely, if null is also written

and

Az

D , then

3 27

Alt m

for some

Y Az ACxtz

null

Ax AZ

AF Ax

y

ya

y

o

AE

AG 3

2

Zero nullspace

AE11210 20

Ia Az

970

is called one-to-one if is the only element of its nullspace

aier

I

y

y fat

null

y Ax Axe

Equivalently,

can always be uniquely determined from

(i.e., the linear transformation

doesn't `lose' information)

O

Alyx

o? mapping from to is one-to-one: different 's map to different 's

columns of are independent (hence, a basis for their span)

has a left inverse, i.e., there is a matrix

s.t.

I

T is invertible

ATAm EIR

nxm

x op

null A I

3

Zero nullspace

BA I

Ax o BAX B O

Ae 1210 20

O

XO

AB

IFE

if has a left inverse then null

null

null T

(proof by contradiction)

ATA

BA ALTAS'ATI I

if null

then is left invertible, because T is invertible, so

T

T is a left inverse

Xe null A Xe null ATA

Ax o

ATA X

BI

ATAx o

xenull ATA

null talc null ATA

O TATA X O LAX TAX o D IA112 0

o

Ax 0

UTU LIMP

AT

penna

NUMATA Chula 4

Two interpretations of nullspace

suppose null , and

represents measurement of

is undetectable from sensors -- get zero sensor readings

and

are indistinguishable from sensors:

null characterizes ambiguity in from measurement

alternatively, if

represents output resulting from input

is an input with no result

and

have same result

null characterizes freedom of input choice for given result

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download