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$$\subseteq$$Y. Show that Y$$\cap$$(X+Z) = X + (Y$$\cap$$Z). (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$$\subseteq$$Y. Show that Y$$\cap$$(X+Z) = X + (Y$$\cap$$Z). (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 $Y\cap (X+ Z)$?