## s3a 2 years ago 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?

1. ingenuus

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

2. s3a

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

3. ingenuus

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

4. s3a

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

5. ingenuus

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

6. ingenuus

it is just a fact for every matrices

7. ingenuus

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

8. s3a

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

9. s3a

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

10. s3a

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

11. s3a

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

12. ingenuus

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

13. ingenuus

matrix is 3by 5 so n=5

14. s3a

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

15. s3a

dim N(A) = n - r that is

16. ingenuus

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

17. ingenuus

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

18. s3a

(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.)

19. s3a

(My break will be when I sleep.)

20. ingenuus

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

21. ingenuus

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

22. s3a

you find the vector x in Ax = 0

23. s3a

?

24. ingenuus

yeah i'm asking do you know the exact procedures

25. s3a

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

26. s3a

(on the left)

27. s3a

i mean left multiplication on each side of the equation

28. ingenuus

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

29. ingenuus

you can do that only when a is invertible

30. ingenuus

you use elimination process to compute nullspace right?

31. ingenuus

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

32. s3a

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?

33. ingenuus

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

34. ingenuus

it is just whole solutions to Ax=0

35. ingenuus

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

36. s3a

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

37. ingenuus

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

38. ingenuus

no nullspace is just a name of subspace

39. ingenuus

dim N(A) is what you are saying

40. s3a

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

41. s3a

Will you be here in 1..75 hours?

42. ingenuus

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

43. s3a

I have to go eat now.

44. ingenuus

i'm sorry

45. s3a

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

46. ingenuus

i have to go :(

47. s3a

o :(

48. ingenuus

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

49. ingenuus

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

50. s3a

thanks and sry for leaving abruptbly