Thread: Help on some Linear Algebra - subspaces

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?

Originally Posted by Lord Voldemort

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?

If the vectors in V and W are subspaces of R^n, the union of of course will be subspace.

The intersection of V and W could be empty. Is an empty set a subspace? I am not too sure of that.

Thanks!
V intersect W is not necessarily a subspace, since the empty set does not contain 0.

Originally Posted by Lord Voldemort

Thanks!
V intersect W is not necessarily a subspace, since the empty set does not contain 0.

I was to lazy to look up anything so goodjob.

If V and W are both subspaces, they both contain the zero vector, and hence the intersection is nonempty.

Originally Posted by dwsmith

If the vectors in V and W are subspaces of R^n, the union of of course will be subspace.

The intersection of V and W could be empty. Is an empty set a subspace? I am not too sure of that.

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.

Originally Posted by TheEmptySet

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.

Originally Posted by TheEmptySet

What is true is if you take the sum this is a subspace.

It's also interesting to note that this is not strange since, if is the ambient space,