The dimension of the intersection of two matrices ranges.

[GoodSpirit]

Hello everyone,
I’d like to ask the following:
Considering the matrices A and B, both with the dimensions \(\displaystyle n \times m\). The ranges are R(A) and R(B). I would like to find.
\(\displaystyle
D= dim( R(A) \cap R(B))
\)
expressed as a function of the dimension of the orthonormal basis of the intersection subspace of R(A) and R(B).
Can you help me?

I sincerely thank you!

All the best

GoodSpirit

 

[HallsofIvy]

"as a function of the orthonormal basis of the intersection subspace. :
If you know an "orthonormal basis of the intersection subspace, the dimension is just the number of vectors in the basis. But that's obvious. Perhaps you meant a basis for the range of each matrix?

 

[GoodSpirit]

Hello, Thank you for your answer!

Well actually what I meant was as function of the dimension of the intersection space of R(A) and R(B)

It is more correct this way

Thank you again

GoodSpirit