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.
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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.
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
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 $\displaystyle W_1$.
