What are the properties of subspaces?

[matrix_204]

I had a question regarding subspaces.
Given vectors (a,b,c,d) s.t. \left{\mid}\begin{array}{cc}a&b\\c&d\end{array}\right{\mid}=0
a supspace of \Re^4?

Though i kno the answer is yes, but i don't understand like it looks to me that it uses one of the three properties of subspaces; the zero vector one i m guessing. But could someone please explain this to me?

 

[Hurkyl]

Don't panic! Stop, take a breath, and collect your thoughts.

You seem to be asking for help in understanding why this particular class of vectors form a subspace. So, you should look at each of the conditions that a subset be a subspace one at a time and see if they apply.

P.S. I think you were trying to write

<br /> \left|<br /> \begin{array}{cc}<br /> a&amp;b \\<br /> c&amp;d<br /> \end{array}<br /> \right|<br />

 

[matrix_204]

lets say v1=(1,0,0,0) and v2=(1,1,1,1), by adding the two vectors, u get (2,1,1,1) and it turns out 2(1)-1(1)doesn't equal 0, so condition one fails, right?

 

[Hurkyl]

If the set is not closed under addition^*, then it is not a subspace.



* don't refer to this simply by number -- the only people who will understand you are those who know your textbook and will look it up

 

[matrix_204]

so it seems like it is not closed under addition, but the teacher said as a hint that it is a subspace of R^4.

 

[Hurkyl]

Well, if you're confident in your analysis, then one of two things must be true:

(a) Your teacher was wrong.
(b) You have misunderstood the problem.


I'll save you a little bit of headache -- if I understand your initial post, then you have correctly analyzed the problem you stated.

 

[matrix_204]

ok i m pretty sure i m wrong, but i will restate the problem in the way the teacher said.
Is the set of all vectors (a,b,c,d) such that \left|\begin{array}{cc}a&amp;b \\c&amp;d\end{array}\right|=0 a subspace of R^4?

 

[Hurkyl]

Have you produced two vectors in that set whose sum is not in that set?
Is that sufficient to prove the set is not a vector space?

 

[matrix_204]

well now i understand a bit more, and checked that there is no vector whose sum is not in the set. So it satisfies the condition of addition. Similarly satisfies the second condition of multiplication, but now I m confused in how to check it using the Zero vector condition. Can you give me an example of checking something using the zero vectors?

 

[Hurkyl]

You sure? What was wrong with your example of:

<1, 0, 0, 0> + <1, 1, 1, 1> = <2, 1, 1, 1>


Anyways, I'm not sure what you mean by "zero vector condition" -- as I remember, the only thing you need to check is closure under addition and multiplication.

Note that closure under multiplication proves that it contains the zero vector (multiply by 0).