Linear Algebra - Basis

[rad0786]

Hello... I am doing this problem with basis. Infact, I am having a lot of problems understanding basis, i did every question in the textbook and I still get seem to understand the idea of it.

So i was hoping somebody can help me with the whole idea about it.

Say, for example, how would i go abouts a question like this:

--Let F be a field and let V = F^3. Let
W = {(a1 a2 a3) E F^3 / 2a1 - a2 - a3 = 0 }

Find a basis for W --

If a question like that came on a test, id fail it - sad to say. It would also be good if somebody knows a good website or has sameple tests that covers this mataril so that i may get used to it.

Thanks

 

[Brad Barker]

so any element in W can be represented like so:

w = (a1, a2, a3), where a1, a2, and a3 are arbitrary.

but W has the additional restriction that a1 = 1/2 (a2 + a3).

so

w= ( 1/2 (a2+a3), a2, a3).

w = a2 ( 1/2, 1, 0) + a3 (1/2, 0, 1). (it's easy to see that this is the same as above.)

so

w = span{(1/2, 1, 0), (1/2, 0, 1)}.


and that set {(1/2, 1, 0), (1/2, 0, 1)} is our basis.


ah, i miss these problems!

 

[Gokul43201]

Keep in mind that the above is not the only basis.

Also, notice that the given vector space is nothing but a (generalization of a) plane through the origin in $\mathbb{R}^3$. Any pair of vectors in the plane will serve as a basis.

 

[rad0786]

Hey.. i was just wondering about Brad Barkers post above...

he said that a1 = 1/2 (a2 + a3).

well.. shouldn't it be a1 = 1/2 (a2 - a3) ?

Does it make a difference?

 

[Gokul43201]

No, you said "2a1 - a2 -a3 = 0"

That gives 2a1 = a2 + a3

 

[rad0786]

oh that was my silly mistake.. but either way.. i still learned something :)

 

[rad0786]

Hey... what about the subspace...

U = (a+b+c=0/ a b c is in the Real Numbers)

How would you show the span of that.

Also, the comment Gokul43201 made, about the basis being the plane through the origin, how did he know that? I mean its a plane because of the two vectors, but how did he know that its through the origin?

 

[iNCREDiBLE]

Also, the comment Gokul43201 made, about the basis being the plane through the origin, how did he know that? I mean its a plane because of the two vectors, but how did he know that its through the origin?

The general equation of a plane is $Ax+By+Cz = D$.
$D = 0 \Longleftrightarrow$ the plane goes through the origin.

 

[rad0786]

"and that set {(1/2, 1, 0), (1/2, 0, 1)} is our basis." Can that be a Basis for F^3? what is F (Feild)? isn't that the same as R, like R^3

 

[Gokul43201]

The general equation of a plane is $Ax+By+Cz = D$.
$D = 0 \Longleftrightarrow$ the plane goes through the origin.

Also, for a plane to constitute a vector space with the usual vector addition, it must pass through the origin, since this point is the additive identity element.