(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let V be a finite vector space, and A, B be any two linear operator. Prove that,

rank A = rank B + dim(Im A [tex]\cap[/tex] Ker B)

3. The attempt at a solution

Since rank A = dim Im A

dim(Im B)+ dim(Ker B)=dim V

dim(Im A + Ker B)=dim(Im A)+dim(Ker B)-dim(Im A [tex]\cap[/tex] Ker B)

It seems equivalent to prove that dim(Im A + Ker B)=dim V

it is obvious that Im A + Ker B [tex]\subseteq[/tex] V. So it should be true that V [tex]\subseteq[/tex] Im A + Ker B.

If I let Ker B=<e_1,...,e_k> and extend it to a basis of V=<e_1,..e_k, e_k+1,...e_n>. Choose any [tex]x=\sum^{n}_{i=1} x_{i}e_{i}=\sum^{k}_{i=1} x_{i}e_{i} + \sum^{n}_{i=k+1} x_{i}e_{i}[/tex]. The first term is in Ker B, the second is not in Ker B and therefore should be in Im A. This is what confuses me a lot since it is not necessary to be true IMO...

What's your idea? Where I made the mistakes? Can you give me any hint to prove it? Thanks very much

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: An equation related to the dimension of linear operator

**Physics Forums | Science Articles, Homework Help, Discussion**