Vector spaces, subspaces, subsets, intersections

  • Thread starter karnten07
  • Start date
1. The problem statement, all variables and given/known data
Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X[tex]\subseteq[/tex]Y. Show that Y[tex]\cap[/tex](X+Z) = X + (Y[tex]\cap[/tex]Z). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)


2. Relevant equations



3. The attempt at a solution

Can anyone get me started on this one?
 

CompuChip

Science Advisor
Homework Helper
4,282
47
Suppose that you have an element [itex]p \in Y \cap (X + Z)[/itex]. Then [itex] p \in Y[/itex] and [itex]p \in X + Z[/itex]. The latter means we can write [itex]p = x + z[/itex] with [itex]x \in X, z \in Z[/itex]. Now do you see how you can also write it as [itex]x' + y'[/itex] with [itex]x' \in X, y' \in Y \cap Z[/itex]?
 

HallsofIvy

Science Advisor
41,620
821
1. The problem statement, all variables and given/known data
Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X[tex]\subseteq[/tex]Y. Show that Y[tex]\cap[/tex](X+Z) = X + (Y[tex]\cap[/tex]Z). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)


2. Relevant equations



3. The attempt at a solution

Can anyone get me started on this one?
The hint looks like all you need. Have you tried that at all? Suppose v is in [itex]Y\cap(X+Z)[/itex]. That means it is in y and it can be written v= au+ bw where u is in X and w is in Z. Now you need to show that v is in [itex]X+ (Y\cap Z)[/itex]. That is, that it can be written in the form au+ bw where u is in X and w is in [itex]Y\cap Z[/itex]. Once you have done that turn it around: if v is in [itex]X+ (Y\cap Z)[/itex], can you show that it must be in [itex]Y\cap (X+ Z)[/itex]?
 

Want to reply to this thread?

"Vector spaces, subspaces, subsets, intersections" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top