If A is an m x n matrix, show that null(A) = [row(A)]_|_ (meaning rowA perp).
I handed in an assignment with this question on it, and got zero points. I think what I did is mostly right, but I want someone to make sure I'm not out to lunch before I go to my prof.
The Attempt at a Solution
The dim[null(A)] = n -r
dim [row(A)] = r
So combining the above equations, we get dim[null(A)] + dim[row(A)] = n
Both null(A) and row(A) are subspaces of Rn. So I can use the theorem that dim(U) + dim(U perp) = n.
Then I wrote that by the above condition, null(A) = (rowA) perp.
And I got zero points out of eight. I think I should have included this line:
U = null(A)
U perp = row(A).
But (rowA) perp = U perp perp = U = null(A), which proves it. In the previous question in the assignment I proved that U perp perp = U, so I don't need to show it again.
This is worth 20 % of my final grade - do you think I should approach my prof about this? My TA was the one who marked it, but she told me what I was doing was wrong. So I'm not sure... thanks!