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!

Printable View

- January 6th 2013, 09:32 AMStormeyTricky 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! - January 6th 2013, 11:19 AMDevenoRe: Tricky question, subspaces dimensions
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 - January 6th 2013, 11:52 AMjohngRe: Tricky question, subspaces dimensions
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.)

- January 6th 2013, 12:14 PMStormeyRe: Tricky question, subspaces dimensions
thanks man!

when i used Venn Diagram i always considered as being equal to . - January 6th 2013, 12:40 PMDevenoRe: Tricky question, subspaces dimensions