Linearly independent vectors and span

In summary: Suppose that we have a vector space with n vectors. We can form a basis for that space by taking n linearly independent vectors and putting them together into a column vector. This basis will span the space because every vector in the space can be written as a linear combination of the vectors in the basis.The basis is also linearly independent because any two vectors in the basis can be written as a linear combination of the two vectors in the basis. In other words, if we take the dot product of two vectors in the basis, we will always get zero. This means that the vectors in the basis are linearly independent.Now suppose that we take away one of the vectors in the basis. This will mean that
  • #1
jasoqueso
1
0
So the question is...

Let v1, v2,...,,vn be linearly independent vectors in a vector space V. Show that v2,...,vn cannot span V.

I honestly have found myself completely lost lately and I suck at writing proofs.

So this is what I see,

v1, v2,..., vn is linearly independent iff c1v1+c2v2+ ... + cnvn = 0 and c1=c2=...=cn= 0

however I don't see why taking away v1 would make it not span V anymore.
 
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  • #2
jasoqueso said:
So the question is...

Let v1, v2,...,,vn be linearly independent vectors in a vector space V. Show that v2,...,vn cannot span V.

I honestly have found myself completely lost lately and I suck at writing proofs.

So this is what I see,

v1, v2,..., vn is linearly independent iff c1v1+c2v2+ ... + cnvn = 0 and c1=c2=...=cn= 0

however I don't see why taking away v1 would make it not span V anymore.

Consider this. How do you write v1 as a linear combination of v2, ..., vn?

Or for a concrete example, in R^3. Take three vectors at right angles called x, y, and z. Can you write x as a linear combination of y and z? Obviously not, but go back to your definition of "span" and "linearly independent" and figure out why!
 
  • #3
Do the following:

1. Re-state the definition of linear independence carefully. (Your language was unclear.)
2. Assume that you can express v1 as a linear combination of the other vectors.
3. Show that this will contradict linear independence.
4. The conclusion is that you cannot express v1 using the other vectors.

It's very important to know your definitions precisely!
 
  • #4
In an n-dimensional vector space, a linearly independent set of n vectors is a basis for that space. Also, in an n-dimensional vector space, every basis for that space has the same amount of vectors. A basis is a linearly independent set that spans the space. This makes the rest of the proof easy.
 
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1. What does it mean for vectors to be linearly independent?

Linearly independent vectors are a set of vectors that cannot be written as a linear combination of each other. This means that no vector in the set can be expressed as a sum of scalar multiples of the other vectors in the set.

2. How can I determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if and only if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 (where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set) is c1 = c2 = ... = cn = 0. In other words, the only way for the linear combination of the vectors to equal the zero vector is if all of the scalars are equal to zero.

3. What is the span of a set of vectors?

The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be expressed as a linear combination of the original set of vectors.

4. How do I determine if a vector is in the span of a set of vectors?

A vector v is in the span of a set of vectors if and only if there exist scalars c1, c2, ..., cn such that c1v1 + c2v2 + ... + cnvn = v, where v1, v2, ..., vn are the vectors in the original set.

5. Can a set of linearly independent vectors have more than one spanning set?

No, a set of linearly independent vectors can only have one spanning set. This is because the span of a set of vectors is unique and is determined by the vectors in the set. If the vectors in the set are linearly independent, then there is only one way to create linear combinations of those vectors, resulting in only one possible spanning set.

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