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.

Printable View

- Jun 7th 2013, 04:58 AMTweetysubspace 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 AMswarnaRe: 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 AMHallsofIvyRe: 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 $\displaystyle W_1$.