- #1

Dafe

- 145

- 0

## Homework Statement

Suppose L, M, and N are subspaces of a vector space.

(a)

Show that the equation

[tex] L \cap (M+N) = (L \cap M)+(L \cap N) [/tex]

is not necessarily true.

(b)

Prove that

[tex] L \cap (M+(L \cap N))=(L \cap M) + (L \cap N) [/tex]

## Homework Equations

N/A

## The Attempt at a Solution

(a)

I let

[tex]

\begin{aligned}

M=&\;span\{(0,0),(1,0)\}\\

N=&\;span\{(0,0),(0,1)\}\\

L=&\;span\{(0,0),(1,1)\}

\end{aligned}

[/tex]

Then,

[tex]

\begin{aligned}

M+N=&\;span\{(0,0),(1,0),(0,1),(1,1)\}\\

L \cap (M+N)=&\;span\{(0,0),(1,1)\}\\

L \cap M=&\;span\{(0,0)\}\\

L \cap N=&\;span\{(0,0)\}

\end{aligned}

[/tex]

and the equation is not true.

This in fact leads me to believe that the equation does not hold when [tex]L \subset (M+N) [/tex], because then [tex]L \cap (M+N) = L[/tex] and [tex]L \cap M[/tex] and [tex]L \cap N[/tex] are something else.

I would guess they turn out to be something like [tex]L-L \cap N[/tex] and [tex]L-L \cap M[/tex], respectively..

(b)

[tex] L \cap M + L \cap (L \cap N) = (L \cap M) + (L \cap N) [/tex]

That's all I can come up with on my own.

Any suggestions are appreciated, thanks!