Dimensions of Matrices Range (equalities).

[GoodSpirit]

Hello everyone,

I’d like to find the following range equalities:
Considering the following:
\(\displaystyle
A=B+C \\
A=B.C^T \\
A=[ B^T C^T ]^T
\)
I would like to find the function \(\displaystyle f \) for each equality above.
\(\displaystyle
.\\
dim( R(A) ) = f( R(B) , R(C) )\\
\)
Considere that all matrices have compatible dimensions.
R(X) function means range of the column space expressed in the matrix X.
Can you help me?

I sincerely thank you!

All the best

GoodSpirit

 

[DrPhil]

Hello everyone,

I’d like to find the following range equalities:
Considering the following:
\(\displaystyle
A=B+C \\
A=B.C^T \\
A=[ B^T C^T ]^T
\)
I would like to find the function \(\displaystyle f \) for each equality above.
\(\displaystyle
.\\
dim( R(A) ) = f( R(B) , R(C) )\\
\)
Considere that all matrices have compatible dimensions.
R(X) function means range of the column space expressed in the matrix X.
Can you help me?

I sincerely thank you!

All the best

GoodSpirit

I hope these are three separate questions! If you had shown your work, that would have been more clear. Actually, If you DON'T show your work, we don't know where you are getting stuck.

Are you having trouble getting started? It looks like you are supposed to consider dimensions (m,n) for each of A, B, C, and then make them compatible with the indicated matrix operations. Then see how the range of A relates to the others. For the first one, for instance, the addition operation is only allowed for matrices of the same dimensions. Thus

\(\displaystyle dim( R(A) ) = dim( R(B) ) = dim( R(C) )\)

Let us see how far you have gotten with the others.

 

[GoodSpirit]

Hi DrPhil,

Thank you for answering!
Well I'm stuck precisely in these equations
It is three questions in one. These equalities are very useful for rank equations simplification.

True these matrices have the same dimensions (m,n). But they can have different ranks so the last equation that you wrote is not true.

I've been using linear transformations and set theory to solve.

By the way, I found the function for the first equation.
\(\displaystyle
\dim(R(A))=\dim(R(B))+\dim(R(C))-\dim(R(B) \cap R(C))
\)

I really thank you

GoodSpirit

 

[DrPhil]

True these matrices have the same dimensions (m,n). But they can have different ranks so the last equation that you wrote is not true.

OOPS - I completely ignored the problem of linear dependence!

 

[GoodSpirit]

Hi,

For the third case is

\(\displaystyle
\dim(R(A))=\dim(R(B))+\dim(R(C))-\dim(R(B^T) \cap R(C^T))
\)

It seems that the second case its a little more complicated.

All the best

GoodsSpirit