Linear Algebra... Subspaces

crazyleonard

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Isn't this vector not non-empty because if you substitute x=0 and z=0 then the middle term will be -3.
I'm not 100 percent sure as to what to do to show if a vector is non-empty so I may be completely wrong. Can someone explain this clearly?
My question for this problem though is... is this vector non-empty
 

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Isn't this vector not non-empty because if you substitute x=0 and z=0 then the middle term will be -3.
Your sideways, over-written, fuzzy graphic is difficult to read. I think the content may be as follows:



\(\displaystyle \mbox{2. Determine whether }\, W\, =\, \left\{\, \left(\, \begin{array}{c} x\\x\, -\, 3\\z \end{array}\, \right)\, \bigg|\, x,\, z\, \in\, \mathbb{R}\, \right\}\, \mbox{ is a subspace of }\, \mathbb{R}^3.\)



However, you then ask about proving something about non-empty vectors, rather than proving something about subspaces, so I may be mistaken.

Kindly please reply with corrections and clarifications. When you reply, please explain what you are doing in attempting to answer the question in the exercise. Thank you! ;)
 
Hi! Yes, that is what the question is... sorry about the bad image quality.

When trying to determine whether a W is a subspace of R^3, there are three things to do if I'm not wrong.

Show it is non-empty, that it is closed under vector addition and that it is closed under scalar multiplication.
So I understand how to show that something is closed under vector addition and that it is closed under scalar multiplication, but I'm not sure how to show that it is non-empty.

I know it has to do with zeros and the zero vector, but what would be an example of a vector which is not non-empty.
 
When trying to determine whether a W is a subspace of R^3, there are three things to do if I'm not wrong.

Show it is non-empty, that it is closed under vector addition and that it is closed under scalar multiplication.
Are you supposed to show that the set is non-empty, or are you supposed to show that the zero vector is an element? (The latter shows the former, but the former does not show the latter.)

I know it has to do with zeros and the zero vector, but what would be an example of a vector which is not non-empty.
A set is empty or non-empty of elements within that set. An element of that set is not itself empty or non-empty. The terminology makes no sense within the context of an element of the set.

So I understand how to show that something is closed under vector addition and that it is closed under scalar multiplication, but I'm not sure how to show that it is non-empty.
To show that a set is non-empty, show that a specific element exists within the set. So pick a value for x, pick a value for z, plug the values in, and list the result. This is an element.

But is the set of such elements also a vector space? ;)
 
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