General

wtf the questions you asked are literally the first basic fundamentals you learn in the course

lol

[quote=KawaiiPony]Thanks that helps a lot. Just to clarify, so say for R2 {(1,0),(0,1),(0,3)} , (3,0) does not span R2 because it is just expanding the line (1,0), therefore it does not span?[/quote]

{(1,0),(0,1),(3,0),(0,3)} spans R2. It's just that you only really need (1,0) and (0,1) to describe the R2 space, since you can get (3,0) and (0,3) with combinations of three (1,0)s and three (0,1)s respectively.

If you don't have any vector that can describe the "y" part of a vector, but you have (3,0), then you don't have a spanning set of R2.
That might look like {(3,0), (1,0), (4,0)}.

Just to be clear, {(3,1), (6,2), (9,3)} also doesn't span R2, even though each individual vector has a "y" component, because the vectors in this set are linearly dependent. So you can only describe one single direction with them. You can write (6,2) and (9,3) in terms of (3,1), but you can't write something like (3,3), so you don't cover every vector in R2.

In much the same way, {(1,0),(1,2)} is a spanning set of R2, because the vectors are linearly independent. You can get something like (3,3) with a combo of (3/2) (1,2) and (3/2) (1,0) since you have two directions to work with.

If you were asked to get the set that spans a 1-dimensional subset of R2, then {(3,1), (6,2), (9,3)} would be okay, since it's describing some line in R2.

Sketchtoy for science:

http://sketchtoy.com/64995470 BAD

http://sketchtoy.com/64995471 GOOD (since you can cover everything if you can describe two separate directions)

[quote=xoqtprincessxo]First question

The vectors have to fully describe the vector space. Example, for your usual 2D plane (R2), you might describe it with two vectors x and y, where x is (1,0) and y is (0,1). Now you can use x and y to describe any other vector in the 2D plane, i.e you get (5,6) with five x's and six y's.

The two vectors you choose shouldn't be able to be described in terms of each other. If you have {(0,1), (0,3)} as your spanning set, you can describe everything in a vertical line and nothing horizontally. You'll never get a vector like (5,6), but you can get (0, 9) with a combination of three (0,3) or two (0,3) and three (0,1), etc etc etc.

You can also have a spanning set for R2 like {(1,0), (0,1), (0,3)}. You don't need to be terribly efficient about describing these things. You can't have a spanning set for R2 like {(0,1)}.
Now generalize this jazz yourself.

For the second question, basil thinks I'm drawing boobs in my answer or something and idk what's wrong, so I'm just giving up.[/quote]

Thanks that helps a lot. Just to clarify, so say for R2 {(1,0),(0,1),(0,3)} , (3,0) does not span R2 because it is just expanding the line (1,0), therefore it does not span?

First question

The vectors have to fully describe the vector space. Example, for your usual 2D plane (R2), you might describe it with two vectors x and y, where x is (1,0) and y is (0,1). Now you can use x and y to describe any other vector in the 2D plane, i.e you get (5,6) with five x's and six y's.

The two vectors you choose shouldn't be able to be described in terms of each other. If you have {(0,1), (0,3)} as your spanning set, you can describe everything in a vertical line and nothing horizontally. You'll never get a vector like (5,6), but you can get (0, 9) with a combination of three (0,3) or two (0,3) and three (0,1), etc etc etc.

You can also have a spanning set for R2 like {(1,0), (0,1), (0,3)}. You don't need to be terribly efficient about describing these things. You can't have a spanning set for R2 like {(0,1)}.
Now generalize this jazz yourself.

For the second question, basil thinks I'm drawing boobs in my answer or something and idk what's wrong, so I'm just giving up.

Are you actually doing this in a linear algebra course or are you in diff eq?
I had to learn all about this in my diff eq class I'm currently in and I feel like this kind of material is a little overboard for a diff eq course.

first poster, I have an exam coming up in a few days. Also I did look up the axioms in my book, but they aren't reaching my understanding. So... I don't know why you gotta show off by acting all tough :C

A vector U is in span V if U is a linear combination of any vectors in the space V. So let's say you have two vectors, A and B. If you can somehow prove that U is a linear combination of A and B, and A and B are in span V, then vector U is in span V. I may or may not be wrong.

The set of vectors must be linear independent. Check the determinant.

These concepts are pretty useless overall though. I agree with the previous poster and your post history suggests you are trying to show off?

How about looking at the axioms in your book?

I'm going to call you out. Basil is a site populated mostly by teenagers. Do you really think you're gonna get an answer here? I don't know why you're trying so hard to show off and impress people you don't even know. Bye.