Question

Question:
If A is a 3 by 5 matrix, what information do you have about the nullspace of A?

Answer:
N(A) has dimension at least 2 and at most 5.

My trouble:
How do we know this?

Answer 1

m>=r
m=3 so 0=<r=<3
dim N(A) = n-r=5-r
2<=5-r<=5

Answer 2

You're using the notation where m is the number of rows and r is the rank, right?

Answer 3

yes
m=#rows
n=#columns
r=rank

Answer 4

So m >= r always holds for the nullspace of a matrix?

Answer 5

rank is #pivot rows right?
can it be more than #of rows?

Answer 6

it is just a fact for every matrices

Answer 7

i mean you can't have more pivot 'rows' than rows

Answer 8

No, it cannot. I now get the answer to that sub-question.

Answer 9

I mean my sub-question not part (b). Let me look at 3(b) again

Answer 10

Why is dim N(A) = n-r=5-r?

Answer 11

Also is dim N(A) = dim ("the x variable vector in Ax = 0")?

Answer 12

i don't know whether you've heard the mit ocw lecture or not, but you should know dim N(A) = n -r

Answer 13

matrix is 3by 5 so n=5

Answer 14

I haven't. Is N(A) = n - r something I am supposed to "just memorize"?

Answer 15

dim N(A) = n - r that is

Answer 16

well dim N(A) is number of vectors in basis of nullspace of A

Answer 17

think about the basis of nullspace of A
think how you get those bases

Answer 18

(where n and r are of the matrix A) (Sorry for the potentially dumb question, I'm kind of braindead at the moment but, like I said, I cannot take a break.)

Answer 19

(My break will be when I sleep.)

Answer 20

can you think how you obtain the null space of A?

Answer 21

then it is clear cut that dim N(A) = n-r

Answer 22

you find the vector x in Ax = 0

Answer 23

?

Answer 24

yeah i'm asking do you know the exact procedures

Answer 25

get A^(-1) and multiply both sides?

Answer 26

(on the left)

Answer 27

i mean left multiplication on each side of the equation

Answer 28

since there are r pivot columns, there are n-r free columns and there are n-r special solutions to the null space. And they form basis for nullspace. so dim N(A)=n-r

Answer 29

you can do that only when a is invertible

Answer 30

you use elimination process to compute nullspace right?

Answer 31

why don't you solve some examples to get dimN(A)=n-r
i think you know the definitions

Answer 32

when a is invertible, like you said.

I'm not grasping something fundamental though: what is the nullspace of A? is it x in Ax = 0? In other words, what object's dimension is n - r?

Answer 33

nullspace(A) is a subspace in R^n, which a vector in it satisfies Ax=0

Answer 34

it is just whole solutions to Ax=0

Answer 35

solutions form a subspace, so they are called null'space'

Answer 36

ok so it's a dimension of a vector space in which a vector x makes Ax = 0?

Answer 37

for example, when A is invertible, x is only 0 and the N(A)={0}

Answer 38

no nullspace is just a name of subspace

Answer 39

dim N(A) is what you are saying

Answer 40

ya for two seconds, my brain was thinking about dim N(A).

Answer 41

Will you be here in 1..75 hours?

Answer 42

you need to carry on the elimination yourself to see dim N(A) = n-r

Answer 43

I have to go eat now.

Answer 44

i'm sorry

Answer 45

(If you keep writing, I will come back and read what you said.)

Answer 46

i have to go :(

Answer 47

o :(

Answer 48

i have a pdf about nullspace. do you want it?

Answer 49

http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/solving-ax-0-pivot-variables-special-solutions/

download the lecture summary for that lecture.
it has the just the right information for you.
good luck

Answer 50

thanks and sry for leaving abruptbly