# Linear Algebra - Basis

Hello... im doing this problem with basis. Infact, im having a lot of problems understanding basis, i did every question in the text book 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

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
Staff Emeritus
Gold Member
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.

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

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

Does it make a difference?

Gokul43201
Staff Emeritus
Gold Member
No, you said "2a1 - a2 -a3 = 0"

That gives 2a1 = a2 + a3

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

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?

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.

"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)? isnt that the same as R, like R^3

Gokul43201
Staff Emeritus
The general equation of a plane is $Ax+By+Cz = D$.
$D = 0 \Longleftrightarrow$ the plane goes through the origin.