# subspace proof help

• Jun 7th 2013, 04:58 AM
Tweety
subspace proof help
V is any vector space, \$\displaystyle U = W_{1} U W_{2} \$ where \$\displaystyle W_{1} , W_{2} \$ are subspace of V.

Prove U is not a subspace of the vector space V.
• Jun 7th 2013, 05:31 AM
swarna
Re: subspace proof help
For example, let us consider two subspaces S & T of the vector space R3 where S = {(x,y,z) : y=0 , z=0} T = {(x,y,z) : x=0, z=0}
let a=(1,0,0) belongs to S & b=(0,1,0) belongs to T. But a+b does not belong to SUT
• Jun 7th 2013, 05:56 AM
HallsofIvy
Re: subspace proof help
You can't prove this, It isn't true unless you have restrictions on \$\displaystyle W_1\$ and \$\displaystyle W_2\$. For example, if the vector space is \$\displaystyle R^3\$, \$\displaystyle W_1= \{x, y, 0\}\$, and \$\displaystyle W_2= \{x, 0, 0\}\$ then \$\displaystyle W_1\cap W_2= W_2\$which certainly is a subspace.

What is true is that \$\displaystyle W_1\cap W_2\$ is a subspace if and only if one is a subspace of the other.

In order to prove this, you will have to make use of the fact, which isn't true unless one is NOT a subspace of the other, that there exist a vector u in \$\displaystyle W_1\$ that is not in \$\displaystyle W_2\$ and that there exist a vector v in \$\displaystyle W_2\$ that is not in \$\displaystyle W_1\$.