Finding a Basis for V: Let V=span(v1, v2, v3, v4)

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In summary, the conversation discusses finding a basis for a given span of vectors and determining linear independence. The participants also discuss the use of matrices in comparison to vectors and solving systems of equations to determine linear independence. There is also a clarification regarding the difference between a span and a basis.
  • #1
Shenlong08
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Homework Statement



Suppose we have,

Let V=span(v1, v2, v3, v4). Find a basis for V.
(there are actual vectors given, however I can't exactly write them in an easy to read form)

Homework Equations


N/A


The Attempt at a Solution



My initial thought is, if I require a basis in which all linear combinations of v1, v2, v3, v4 that can be written, couldn't the v1, v2, v3, v4 be the basis so long as they are linearly independent? As well, wouldn't the standard basis vector (this is for matrices) be valid? However if it was that easy...well I'm probably wrong. I don't really need a direct solution; really just a clarification.
 
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  • #2
Well, what's a basis? It's a linearly independent spanning set. So how about you try to discard the linearly dependent vectors from {v1, v2, v3, v4} (i.e. the ones in this set that can be written as a linear combination of the others)? This will certainly make this set linearly independent, but will it stay spanning?
 
  • #3
If V is the span of {v1,v2,v3,v4} and they are linearly independent, then they are a basis. If there are actual vectors given and this is a problem then it's likely that they aren't linearly independent and you are supposed to eliminate the ones that are linear combinations of the others.
 
  • #4
So I should expect my answer to be 3 out of the 4 vectors, or perhaps 2 then? That would make sense.

I also need help with comparing vectors as matrices. With vectors in R space I know what to do, but for matrices I'm not quite sure.

Suppose I have matrices A, B, C, D such that

aA+bB+cC+dD=0, thus I can solve the system and find values of a, b, c, d. Would this be the right approach to find if its linearly independent? If it is, I'm not quite sure how to interpret my results since I get a=0, b=0, c=0, d=t, where t is a free variable. Meaning that there isn't just the trivial solution and its linearly dependent...however how would I go about finding which one I should remove?

Also, technically, is it wrong to say that the span is the standard basis for matrice vectors? Because A, B, C, and D are just subspaces of all matrices right?
 
  • #5
How can you get a solution like that unless D=0?
 
  • #6
A, B, C, and D are matrices, so I get multiple equations in which I'm solving by row reducing the matrix.
 
  • #7
So I'm getting

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 0

Sorry for this double post, I clicked edit and didn't realize where I was typing.
 
  • #8
your basis is
1 0 0 0
0 1 0 0
0 0 1 0

the zero lines says that the 4th vector is composed out of these 3 vectors
 
  • #9
Shenlong08 said:
So I should expect my answer to be 3 out of the 4 vectors, or perhaps 2 then? That would make sense.

I also need help with comparing vectors as matrices. With vectors in R space I know what to do, but for matrices I'm not quite sure.

Suppose I have matrices A, B, C, D such that

aA+bB+cC+dD=0, thus I can solve the system and find values of a, b, c, d. Would this be the right approach to find if its linearly independent? If it is, I'm not quite sure how to interpret my results since I get a=0, b=0, c=0, d=t, where t is a free variable. Meaning that there isn't just the trivial solution and its linearly dependent...however how would I go about finding which one I should remove?

Also, technically, is it wrong to say that the span is the standard basis for matrice vectors? Because A, B, C, and D are just subspaces of all matrices right?
Saying "aA+bB+ cC+ dD= 0" and then "I get a= 0, b= 0, c= 0, d= t, where t is a free variable" means that tD= 0 so either t= 0 (and your matrices are independent) or D= 0. I don't believe either of those is true.

I can't even make sense out of that last paragraph! No, the "span" is NOT "the standard basis". A "span" is a subspace, a "basis" is a collection of independent vectors.
And, since you had already stated that A, B, C, D are matrices they are certainly not "subspaces of matrices".
 

What is a basis for a vector space?

A basis for a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be written as a linear combination of the basis vectors.

How do I find a basis for a given vector space?

To find a basis for a given vector space, you can use the method of Gaussian elimination to reduce the given vectors to their row-echelon form. The non-zero rows in the resulting matrix form a basis for the vector space.

What is the minimum number of vectors needed for a basis?

The minimum number of vectors needed for a basis is equal to the dimension of the vector space. So, for a 3-dimensional vector space, you will need at least 3 linearly independent vectors to form a basis.

Can a vector space have more than one basis?

Yes, a vector space can have infinitely many bases. This is because there are infinitely many ways to choose a set of linearly independent vectors that span the space.

What is the significance of finding a basis for a vector space?

Finding a basis for a vector space is important because it allows us to represent any vector in the space as a linear combination of the basis vectors. This makes it easier to perform calculations and understand the properties of the vector space.

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