Vector spaces, subspaces, subsets, intersections

[karnten07]

Homework Statement

Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X\subseteqY. Show that Y\cap(X+Z) = X + (Y\capZ). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)

Homework Equations

The Attempt at a Solution

Can anyone get me started on this one?

 

[CompuChip]

Suppose that you have an element p \in Y \cap (X + Z). Then p \in Y and p \in X + Z. The latter means we can write p = x + z with x \in X, z \in Z. Now do you see how you can also write it as x' + y' with x' \in X, y' \in Y \cap Z?

 

[HallsofIvy]

Homework Statement

Let V be a vector space over a field F and let X, Y and Z be a subspaces of V such that X\subseteqY. Show that Y\cap(X+Z) = X + (Y\capZ). (Hint. Show that every element of the LHS is contained on the RHS and vice versa.)

Homework Equations

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 Y\cap(X+Z). 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 X+ (Y\cap Z). That is, that it can be written in the form au+ bw where u is in X and w is in Y\cap Z. Once you have done that turn it around: if v is in X+ (Y\cap Z), can you show that it must be in Y\cap (X+ Z)?