Vectors Spanned & Linear Independence

In summary, the conversation discusses the concept of vector span and linear independence. It is stated that a set of vectors span a space when they generate the space, but this does not necessarily mean they form a basis. In order to determine if a vector is within the span of a set of vectors, one must first check if the vectors are linearly dependent. If they are, the vector will be within the span if it is linearly dependent with any two of the vectors in the set.
  • #1
torquerotates
207
0
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?
 
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  • #2
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).
 
  • #3
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.
 
  • #4
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.
 
  • #5
Thanks.
 
  • #6
thanks morphism for a really clear explanation
 
  • #7
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).
 
  • #8
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...]
 
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  • #9
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  • #10
That is, in fact, pretty much the definition of "span" of a set of vectors!
 

Related to Vectors Spanned & Linear Independence

1. What is the definition of "vectors spanned"?

The term "vectors spanned" refers to a set of vectors that, when combined, can form any vector within a given vector space. In other words, the vectors span the entire vector space.

2. How do you determine if a set of vectors is linearly independent?

A set of vectors is considered linearly independent if none of the vectors in the set can be written as a linear combination of the other vectors. This means that no vector in the set can be expressed as a sum of scalar multiples of the other vectors.

3. Can a set of vectors be both linearly independent and spanned?

Yes, it is possible for a set of vectors to be both linearly independent and spanned. This means that the vectors in the set are not linearly dependent on each other, and they can still span the entire vector space.

4. What is the relationship between vectors spanned and linear independence?

The concept of vectors spanned and linear independence are closely related. A set of vectors is considered spanned if they can form any vector within a given vector space. On the other hand, a set of vectors is considered linearly independent if none of the vectors can be written as a linear combination of the others. In other words, a set of vectors must be spanned in order to be linearly independent.

5. How can you determine if a set of vectors spans a specific vector space?

To determine if a set of vectors spans a specific vector space, you can use the following steps:

  1. Write the vectors as columns in a matrix
  2. Perform row reduction on the matrix
  3. If the reduced matrix has a pivot in every row, then the vectors span the vector space

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