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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?
 one year ago
 one year 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?
 one year ago
 one year ago

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ingenuusBest ResponseYou've already chosen the best response.0
m>=r m=3 so 0=<r=<3 dim N(A) = nr=5r 2<=5r<=5
 one year ago

s3aBest ResponseYou've already chosen the best response.0
You're using the notation where m is the number of rows and r is the rank, right?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
yes m=#rows n=#columns r=rank
 one year ago

s3aBest ResponseYou've already chosen the best response.0
So m >= r always holds for the nullspace of a matrix?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
rank is #pivot rows right? can it be more than #of rows?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
it is just a fact for every matrices
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
i mean you can't have more pivot 'rows' than rows
 one year ago

s3aBest ResponseYou've already chosen the best response.0
No, it cannot. I now get the answer to that subquestion.
 one year ago

s3aBest ResponseYou've already chosen the best response.0
I mean my subquestion not part (b). Let me look at 3(b) again
 one year ago

s3aBest ResponseYou've already chosen the best response.0
Why is dim N(A) = nr=5r?
 one year ago

s3aBest ResponseYou've already chosen the best response.0
Also is dim N(A) = dim ("the x variable vector in Ax = 0")?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
i don't know whether you've heard the mit ocw lecture or not, but you should know dim N(A) = n r
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
matrix is 3by 5 so n=5
 one year ago

s3aBest ResponseYou've already chosen the best response.0
I haven't. Is N(A) = n  r something I am supposed to "just memorize"?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
well dim N(A) is number of vectors in basis of nullspace of A
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
think about the basis of nullspace of A think how you get those bases
 one year ago

s3aBest ResponseYou've already chosen the best response.0
(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.)
 one year ago

s3aBest ResponseYou've already chosen the best response.0
(My break will be when I sleep.)
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
can you think how you obtain the null space of A?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
then it is clear cut that dim N(A) = nr
 one year ago

s3aBest ResponseYou've already chosen the best response.0
you find the vector x in Ax = 0
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
yeah i'm asking do you know the exact procedures
 one year ago

s3aBest ResponseYou've already chosen the best response.0
get A^(1) and multiply both sides?
 one year ago

s3aBest ResponseYou've already chosen the best response.0
i mean left multiplication on each side of the equation
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
since there are r pivot columns, there are nr free columns and there are nr special solutions to the null space. And they form basis for nullspace. so dim N(A)=nr
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
you can do that only when a is invertible
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
you use elimination process to compute nullspace right?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
why don't you solve some examples to get dimN(A)=nr i think you know the definitions
 one year ago

s3aBest ResponseYou've already chosen the best response.0
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?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
nullspace(A) is a subspace in R^n, which a vector in it satisfies Ax=0
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
it is just whole solutions to Ax=0
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
solutions form a subspace, so they are called null'space'
 one year ago

s3aBest ResponseYou've already chosen the best response.0
ok so it's a dimension of a vector space in which a vector x makes Ax = 0?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
for example, when A is invertible, x is only 0 and the N(A)={0}
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
no nullspace is just a name of subspace
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
dim N(A) is what you are saying
 one year ago

s3aBest ResponseYou've already chosen the best response.0
ya for two seconds, my brain was thinking about dim N(A).
 one year ago

s3aBest ResponseYou've already chosen the best response.0
Will you be here in 1..75 hours?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
you need to carry on the elimination yourself to see dim N(A) = nr
 one year ago

s3aBest ResponseYou've already chosen the best response.0
(If you keep writing, I will come back and read what you said.)
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
i have a pdf about nullspace. do you want it?
 one year ago

ingenuusBest ResponseYou've already chosen the best response.0
http://ocw.mit.edu/courses/mathematics/1806sclinearalgebrafall2011/axbandthefoursubspaces/solvingax0pivotvariablesspecialsolutions/ download the lecture summary for that lecture. it has the just the right information for you. good luck
 one year ago

s3aBest ResponseYou've already chosen the best response.0
thanks and sry for leaving abruptbly
 one year ago
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