Vectors Spanned & Linear Independence

[torquerotates]

I'm kinda confused about whether the vectors in a linear span has to be independent. It makes sense intuitively. For example say v and u spans a plane. Then v and u has to be linearly independent. Otherwise they would lie in a line. Can anyone give me an example where vectors span a space and yet are not linearly independent?

 

[morphism]

A line is a perfectly valid space.

And of course any two vectors that span a plane are going to be linearly independent, just like any three vectors that span a 3-dimensional space are going to be linearly independent, or more generally, like any n vectors that span an n-dimensional space are going to be linearly independent (but any collection of more than n vectors isn't going to be).

 

[torquerotates]

So if I have (1,0) , (0,1) and (2,7), these vectors would span all R^2? But I was thinking that out of the set, only the unit vectors span R^2. Because (2,7) is just a combination of i and j, it is covered in the span of the unit vectors. Thus the set doesn't really span R^2, just the unit vectors. There must be something wrong with my thinking. Because what you stated is a theorem.

 

[morphism]

I think you're confusing the notion of a list of vectors spanning a space with that of them forming a basis for the space. When they span, that just means they 'generate' the space -- it doesn't mean the list has to be efficient, i.e. without redundant entries. A basis on the other hand is just that: it's the least redundant list of vectors that spans a space, so that if you remove any single vector from the list, it will no longer be spanning.

Hope this clears things up for you.

 

[torquerotates]

Thanks.

 

[froster]

thanks morphism for a really clear explanation

 

[sihag]

in both Euclidean and Unitary spaces of n dimension, n linearly independent vectors will be a minimal spanning set, which can be proved to be a basis in turn.
Another way of looking at is, in an 'n' dimensional space maxminum number of linearly independent vectors is n, and any n linearly independent vectors (maximal linearly independent set) would be a basis for the space (of course spanning it).

 

[ASU NGNER]

wish I could delete earlier post

So how can I find out if a vector "v" in R3 is in the span of three other vectors, making up th columns of matrix "u" in R3?

Proposed solution:
1. determine if the vectors in u are dependent
yes: move on to step 2
no: u spans all R3 therefore any other vector in R3 is within the span

2. if the vectors in u are dependent, they span a plane in R3. Determine if v is linearly dependent WRT any two of the three vectors in u.
yes: the vector is within the span
no: the vector is not in the span

[I wish I could remove that reply from earlier. It was late, my mind was,... well I'm not sure where it was. I was thinking, just stupidly.

Wait I can, sweet

I never made that stupid comment... really...]

 

[ASU NGNER]

I've found my answer, I think. For anyone else who might need it...

From another https://www.physicsforums.com/showthread.php?t=193169&highlight=span":

Q"if any vector in P_3 can be written as a linear combo of the vectors in S, then can i conclude that the set S spans P_3?"

A:"Exactly!"

 

[HallsofIvy]

That is, in fact, pretty much the definition of "span" of a set of vectors!