# Problem????

Discussion in 'Math' started by dogar sahab, Oct 29, 2008.

1. ### dogar sahab Thread Starter Active Member

Mar 15, 2008
116
1
Q-If the product AB is the zero matrix,AB=0 show that the coulumn space of B is contained in the nullspace of A???
HOW??>>

2. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
Do you know what the terms column space and nullspace mean? If you find these for me I'll help you out.

3. ### dogar sahab Thread Starter Active Member

Mar 15, 2008
116
1
I know how to find but this is an abstract question...so its a proof and i am not getting how to do it!!!
I know null space is
Ax=0

4. ### Dave Retired Moderator

Nov 17, 2003
6,960
162
Mark, try http://en.wikipedia.org/wiki/Four_fundamental_subspaces

OP, the trick is to consider the Transpose of A in terms of the dot-product of the vectors A and B. Then look at the conditions for:

$A^{T}B = 0$

Do you notice anything?

Dave

5. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
Dave, I know what the terms mean. I'm trying to get the OP to show me that he knows, too.

6. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
This is not the definition of the null space. If you don't know what the null space and column space mean, you won't be able to do the proof. I guarantee it.

7. ### Dave Retired Moderator

Nov 17, 2003
6,960
162
Sorry, I just thought you were up for a challenge!

Agreed.

Dave

8. ### dogar sahab Thread Starter Active Member

Mar 15, 2008
116
1
I didnt want to get into details of definition..
If u want then
Column space of any matrix is actually the space spanned by the independent coulums of the given matrix!!!The all other colmn are just the linear combinations of the the columns that lie in the column space...So the whole column space generated by the independent columns of a matrix..This is what the basic concept of column space!!!!

9. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
That's roughly it- the column space of a matrix is the space spanned by its columns (you don't need to add "independent").

Now, what is the nullspace of a matrix A?

doesn't convince me that you know what the term null space means.

10. ### dogar sahab Thread Starter Active Member

Mar 15, 2008
116
1
I guess the term INDEPENDENT COULUMS is more accurate because the dependent columns are included in that space as the dependent columns are actually linear combinations of independent columns!!!

11. ### Mark44 Well-Known Member

Nov 26, 2007
626
1
The column space is the space spanned by the columns of a matrix, whether the columns are independent or dependent vectors. If the columns are linearly independent, the the dimension of the space is the same as the number of columns, and the columns will form a basis for the space. If the the columns are linearly dependent, then the dimension of the space will be less than the number of columns.

Did you ever come up with the definition for the nullspace of a matrix?