Linear Algebra Vector Spanning Question

[caels]

I honestly don't even know where to start with this question. The question is:

Give an example of a span of three vectors u, v, w in four dimensions that corresponds to a plane thru the origin?

I know that the span of the vectors is the set of linear combinations of the vectors. I don't understand how to write a span of some vectors that happen to span four dimensions. I looked through all my notes and my linear algebra book and I can't seem to figure it out, so I don't have any work to sure because I can't figure out where to start. Any help would be appreciated.

 

[jbunniii]

How about if you start by choosing a single vector in four dimensions, for example [1 0 0 0]. What is the span of this vector?

 

[80past2]

Well, it goes through a plane right? That means, even though they're four dimensional, they in effect only cover three dimensions. which is why a set of three can span this subspace. (I think, it's been awhile since I've done linear).

 

[caels]

How about if you start by choosing a single vector in four dimensions, for example [1 0 0 0]. What is the span of this vector?

Well, that's just a line - but I don't see how that helps me ?

 

[jbunniii]

Well, that's just a line - but I don't see how that helps me ?

That was just to get you started. Now let's add a second vector: what is the span of these two vectors? {[1 0 0 0], [0 1 0 0]}

 

[caels]

That was just to get you started. Now let's add a second vector: what is the span of these two vectors? {[1 0 0 0], [0 1 0 0]}

They aren't multiplies of each other, so they must be a plane. To find the final vector could I simply say

a [1 0 0 0] + b [0 1 0 0] = x

Pick arbitrary scalars for a and b, since, by definition, x would have to be in the plane formed by a and b?

 

[jbunniii]

They aren't multiplies of each other, so they must be a plane. To find the final vector could I simply say

a [1 0 0 0] + b [0 1 0 0] = x

Pick arbitrary scalars for a and b, since, by definition, x would have to be in the plane formed by a and b?

Yes, that's exactly what you would do. The key is that the third vector must also lie in the plane, which means it must be a linear combination of the first two vectors.

If instead you had picked something like x = [0 0 1 0], then the span of the three vectors would be too large: a 3-dimensional space instead of a plane.

 

[caels]

Yes, that's exactly what you would do. The key is that the third vector must also lie in the plane, which means it must be a linear combination of the first two vectors.

If instead you had picked something like x = [0 0 1 0], then the span of the three vectors would be too large: a 3-dimensional space instead of a plane.

Thanks for the help. I get it now.