Linear Algebra- Spanning Sets definition

[Roni1985]

Linear Algebra- Basis

Homework Statement

Is {e1,e2} a basis for R³ ?

Homework Equations

The Attempt at a Solution

I know that {e1,e2,e3} is a basis for R³

same here,

Is this one a basis for R³
{(1,1,2)^T,(2,2,5)^T}

I know that
{(1,1,2)^T,(2,2,5)^T,(3,4,1)^T}
is a basis for R³.

Perhaps, I am a little unclear with the definition.

Would appreciate any help.

Roni

 

[Tinyboss]

If you know that {e1,e2,e3} is a basis, then, by the definition of a basis...what can you say about {e1,e2}?

 

[Mark44]

Following Tinyboss's lead, what is the definition of a basis?

 

[Roni1985]

If you know that {e1,e2,e3} is a basis, then, by the definition of a basis...what can you say about {e1,e2}?

This is the part I don't understand...

 

[Roni1985]

Following Tinyboss's lead, what is the definition of a basis?

the vectors must be independent and they must span R^3

but say I have {e1,e2} only. they are independent and they span R^3 (they do span R^3, right?)

 

[Mark44]

They are linearly independent but there are only two of them. How many vectors must there be in a basis for R^3?

 

[Roni1985]

They are linearly independent but there are only two of them. How many vectors must there be in a basis for R^3?

Well, the logic says 3, but I don't understand why 3 ?

and I guess if it's there {e1,e2} is missing 1 vector.
but why 3 ?
here is another example {(1,1,2)^T,(2,2,5)^T}

Thanks for your help.

 

[Mark44]

The number of vectors in a basis for a vector space is equal to the dimension of the vector space. For R^1, any basis must have 1 vector, for R^2, any basis must have two vectors, and so on. If a set contains fewer vectors than the dimension of the vector space it needs to span, it can't possibly span the space, hence can't be a basis.

With your first example, where e1 = (1, 0, 0)^T and e2 = (0, 1, 0)^T, is there some linear combination of these vectors that generates (0, 2, 5)^T?

 

[Roni1985]

The number of vectors in a basis for a vector space is equal to the dimension of the vector space. For R^1, any basis must have 1 vector, for R^2, any basis must have two vectors, and so on. If a set contains fewer vectors than the dimension of the vector space it needs to span, it can't possibly span the space, hence can't be a basis.

With your first example, where e1 = (1, 0, 0)^T and e2 = (0, 1, 0)^T, is there some linear combination of these vectors that generates (0, 2, 5)^T?

I think I understand it...
I guess I'll need to sit on it a little more.

Thank you very much for your help:)

 

[Roni1985]

If a set contains fewer vectors than the dimension of the vector space it needs to span, it can't possibly span the space, hence can't be a basis.

Now this part is confusing me.

So,
{e1,e2} can't span R^3 ?

Oh they must span R^2 with two rows only, right?

what about {(1,1,2)^T,(2,2,5)^T}, what do they span ? R^3?! ( they have two vectors) , R^2 ?! (they have 3 rows so they can't really span R^2, right?!)

Thanks.

 

[jbunniii]

Now this part is confusing me.

So,
{e1,e2} can't span R^3 ?

Oh they must span R^2 with two rows only, right?

Not exactly. In this context, e1 and e2 are elements of R^3, so they are NOT elements of R^2 and therefore cannot span R^2. They do span something that is very similar to R^2, though. Can you see what it is?

what about {(1,1,2)^T,(2,2,5)^T}, what do they span ? R^3?! ( they have two vectors) , R^2 ?! (they have 3 rows so they can't really span R^2, right?!)

Same deal here, the vectors are in R^3, so they aren't in R^2 and can't span R^2. Have you learned what a "subspace" is yet?

 

[Roni1985]

Not exactly. In this context, e1 and e2 are elements of R^3, so they are NOT elements of R^2 and therefore cannot span R^2. They do span something that is very similar to R^2, though. Can you see what it is?



Same deal here, the vectors are in R^3, so they aren't in R^2 and can't span R^2. Have you learned what a "subspace" is yet?

We studied all this, but the professor wasn't really helpful, so everything is on me...
Let me see if I undertand.
{e1,e2} does span R^3 but it doesn't include ALL vectors in the plane that's created.
However,
{e1,e2,e3} also spans R^3 but it forms a solid which is spread throughout all R^3, therefore, it includes ALL vectors in R^3

and {e1} also spans R^3 but it forms a line and any combination of it must be on the line. However, there lots of vectors that are not on the line, therefore, this one cat be a basis for R^3 either.

Right?

Thanks a lot for your help.

 

[jbunniii]

We studied all this, but the professor wasn't really helpful, so everything is on me...
Let me see if I undertand.
{e1,e2} does span R^3 but it doesn't include ALL vectors in the plane that's created.

No, it does not span R^3. You need at least three vectors to do that.

{e1,e2} spans a 2-dimensional plane (subspace) contained within R^3.

Think of the x-y plane in Euclidean 3-space. That plane is spanned by two vectors, for example, the unit vectors in the x- and y-directions. The coordinates of these vectors are (1,0,0) and (0,1,0). This plane is not the same as R^2, where the vectors have only two coordinates. But it's equivalent in a certain mathematical sense (we say that the x-y plane within R^3 is ISOMORPHIC to R^2).

However,
{e1,e2,e3} also spans R^3 but it forms a solid which is spread throughout all R^3, therefore, it includes ALL vectors in R^3

This is what is meant by "spanning" R^3. ANY vector in R^3 can be formed by an appropriate linear combination of e1, e2, and e2. This is NOT true if you only use e1 and e2, and thus {e1,e2} does not span R^3.

and {e1} also spans R^3

No. It spans a one-dimensional line (subspace) contained in R^3. That line is isomorphic to R^1.

but it forms a line and any combination of it must be on the line. However, there lots of vectors that are not on the line, therefore, this one cat be a basis for R^3 either.

Right, it's not a basis for R^3, BECAUSE it does not span R^3.

There are two things that must be true in order for a set of vectors to be a basis for a vector space:

(1) the vectors must span the whole space
(2) the vectors must be linearly independent.

{e1,e2,e2} are linearly independent, and so are any subset of them, so (2) is not the problem here. But in order to achieve (1) you need to have enough vectors. R^3 has dimension 3, and that means you need at least 3 vectors to span it. Furthermore, you can't have MORE than 3 linearly independent vectors in a 3-dimensional space, so if the vectors are to form a basis (a stronger condition than simply spanning the space) you need to have exactly three (linearly independent) vectors.

 

[Roni1985]

No, it does not span R^3. You need at least three vectors to do that.

{e1,e2} spans a 2-dimensional plane (subspace) contained within R^3.

Think of the x-y plane in Euclidean 3-space. That plane is spanned by two vectors, for example, the unit vectors in the x- and y-directions. The coordinates of these vectors are (1,0,0) and (0,1,0). This plane is not the same as R^2, where the vectors have only two coordinates. But it's equivalent in a certain mathematical sense (we say that the x-y plane within R^3 is ISOMORPHIC to R^2).



This is what is meant by "spanning" R^3. ANY vector in R^3 can be formed by an appropriate linear combination of e1, e2, and e2. This is NOT true if you only use e1 and e2, and thus {e1,e2} does not span R^3.



No. It spans a one-dimensional line (subspace) contained in R^3. That line is isomorphic to R^1.



Right, it's not a basis for R^3, BECAUSE it does not span R^3.

There are two things that must be true in order for a set of vectors to be a basis for a vector space:

(1) the vectors must span the whole space
(2) the vectors must be linearly independent.

{e1,e2,e2} are linearly independent, and so are any subset of them, so (2) is not the problem here. But in order to achieve (1) you need to have enough vectors. R^3 has dimension 3, and that means you need at least 3 vectors to span it. Furthermore, you can't have MORE than 3 linearly independent vectors in a 3-dimensional space, so if the vectors are to form a basis (a stronger condition than simply spanning the space) you need to have exactly three (linearly independent) vectors.

Oh, I think I got it...
{e1,e2} span some subspace which is part of R^3, but it doesn't span R^3 because I can't create any vector in R^3 with the linear combination of e1 and e2.

Thank you so much for your help.