i need to disprove the following with an example:
Let be subspaces of .
i used Venn Diagram and covered any possible variations of and but nothing seems to work.
Thanks in advanced!
suppose V = R^{3}, and set:
U_{1} = span({(1,1,0)})
U_{2} = span({(1,0,0)})
U_{3} = span({(0,1,0)}).
then dim(U_{1}+U_{2}+U_{3}) = 2
but dim(U_{1}) + dim(U_{2}) + dim(U_{3}) - dim(U_{1}∩U_{2}) - dim(U_{1}∩U_{3}) - dim(U_{2}∩U_{3}) + dim(U_{1}∩U_{2}∩U_{3})
= 1 + 1 + 1 - 0 - 0 - 0 + 0 = 3
Presumably, this question arose from the true statement that . When you try to extend this to 3 subspaces, you need to find , which is definitely not equal to . Basically, this is why Deveno's counter example above works. Notice though if or , then , and so in this case your original statement is true. (It boils down to the true statement about two subspaces.)