Thread: Tricky question, subspaces dimensions

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 = R³, and set:

U₁ = span({(1,1,0)})

U₂ = span({(1,0,0)})

U₃ = span({(0,1,0)}).

then dim(U₁+U₂+U₃) = 2

but dim(U₁) + dim(U₂) + dim(U₃) - dim(U₁∩U₂) - dim(U₁∩U₃) - dim(U₂∩U₃) + dim(U₁∩U₂∩U₃)

= 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.)

thanks man!

when i used Venn Diagram i always considered as being equal to .

Originally Posted by johng

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.)

exactly so: we might have and , but that does not guarantee that .