Can Rank Multiplication Help Prove a Linear System Statement?

[annoymage]

Homework Statement

Show that rank(A+B) $$\leq$$ rank(A) + rank (B)

for every A,B $$\in$$ M_m,n (Real)

Homework Equations

N/A

The Attempt at a Solution

i only know how to proof this

rank(AB) $$\leq$$ rank(A) or rank(B),



and can this "rank(AB) $$\leq$$ rank(A) or rank(B)" help me to prove the above statement? can someone help me, to prove the above statement

 

[irycio]

I'm not really sure whether it's a fine proof, so please correct me. And forgive my English, never read anything abt algebra in English :).

Let A and B be nxm matrices. Let's consider their columns as the columns of vector coordinates over the same vector space. Now rank(A) and rank(B) are the numbers of lineary independent vectors in each matrix respectively. Adding those two matrices, you are adding vectors. As there can be some linear dependencies between vectors in A and B, rk(A+B) can not overcome rk(A)+rk(B), as it still is nxm matrix. Basically speaking, by taking linear combinations of vectors, you can not get an independent vector, according to the very deffinition itself.

Huh, I hope one can understand what I wanted to write ;).

 

[annoymage]

thanks i get it, and i will try convert that to mathematical form,

thank you

 

[lanedance]

yep, good thinking