Homework SourseGiven U and W subspaces of a vector space V Is the union U
Given U and W subspaces of a vector space V . Is the union U W a subspace of V ? Why or why not?
Solution
The reason why this can happen is that all vector spaces, and hence subspaces too, must be closed under addition and scalar multiplication. The union of two subspaces takes all the elements already in those spaces, and nothing more. To prove that U W need not be a subspace of V, a counter example will suffice. In the union of subspaces U and W, there are new combinations of vectors we can add together that we couldn\'t before, like u1 +w2 where u1 U and w2 W.
For example, take U to be the x-axis and W to be the y-axis, both subspaces of R2. Their union includes both (3,0) and (0,5), whose sum, (3,5), is not in the union U W . Hence, the union U W is not a vector space.
| | The reason why this can happen is that all vector spaces, and hence subspaces too, must be closed under addition and scalar multiplication. The union of two subspaces takes all the elements already in those spaces, and nothing more. To prove that U W need not be a subspace of V, a counter example will suffice. In the union of subspaces U and W, there are new combinations of vectors we can add together that we couldn\'t before, like u1 +w2 where u1 U and w2 W. For example, take U to be the x-axis and W to be the y-axis, both subspaces of R2. Their union includes both (3,0) and (0,5), whose sum, (3,5), is not in the union U W . Hence, the union U W is not a vector space. |
