Mainly I just want verification if this is correct or otherwise.
Consider two subspaces V and W of R_n.
Is V U W necessarily a subspace of R? Is the intersect of V and W a subspace?
Yes to both?
If a vector x is an element of V intersect W, then it is an element of V, then therefore all linear combinations are also in V. It is also in W, and therefore all linear combinations are in W as well. Thus all linear combinations are in V intersect W. The other one would use similar logic.
ALSO
Consider vectors v1, v2... v_m in |Rn. Is the span of the vectors necessarily a subspace of Rn?
I think this is true but how would I justify it?
Say that span is all linear combinations, and then c can be 0 so that fulfills one requirement.
Then say that all scalar multiples of any vectors with real numbers will result in vectors of real numbers?
I have to disagee. The union of two subspaces is, NOT in general, a subspace. Here is a very simple counter example.
Lets consider
Now consider the subspace spanned by (the x axis)
and the subspace spanned by (the y axis)
Now the Union of these two subspaces is both coordinate axes. This is not a vector space at
What is true is if you take the sum this is a subspace.
In fact, if then if and only if or . Indeed, suppose neither containment held, then we may choose and . A quick check then shows that since otherwise contrary to construction. Similarly, . Thus, etc. In fact, this is really just a theorem about the underlying group structure of the vector spaces.