How to extend a set to form a basis?

In summary, the conversation discusses extending a set to be a basis for a vector space, with the general method being to find the span of the original set and then add linearly independent vectors until a basis is formed. The possibility of finding a better or easier method is also discussed, with suggestions of guessing and checking or using the fact that a matrix with linearly independent vectors is similar to the identity matrix. The conversation also mentions the potential for a case-by-case approach.
  • #1
mitch_1211
99
1
i want to extend the set S={(1,1,0,0),(1,0,1,0)} to be a basis for R4. I know I am going to need 4 vectors, so i need to find 2 more that aren't linear combinations of the first 2.

Is there a better way to approach this other than choose 2 at random and check linear independence/dependence until i find 2 that work?

thanks!
 
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  • #2
Guessing is actually a decent way of doing this problem. It's late, but it seems to me that your two other vectors could be {0,0,1,1},{0,1,0,1}. Try to see if this is a basis for R4. Off the top of my head this seems about right.
 
  • #3
In your particular case, a vectort with a nonzero fourth coordinate would work, but
in general, let me think.
 
  • #4
There is no particular method.
Yes, choosing wisely for some vectors that look like to be independent, and then computing the determinant or the rank to check if they are really so.
 
  • #5
In general to extend a set to a basis, you find the span of your original set. Pick any vector not in the span (this guarantees linear independence) and use this as a new basis vector, adding it as a member to your original set. Take the span of this new set and use the same process. Rinse and repeat until you've found a basis. (This process will always terminate for finite-dimensional vector spaces.)

In other words, given a vector space [itex] V [/itex] and a set [itex] S = \{v_1, \ldots, v_n\} [/itex], if S doesn't span V, then [itex] V \backslash \text{span}(S)[/itex] is nonempty, so we may choose a linearly independent vector [itex]w_1[/itex] from this set. Form [itex]S_1 = \{v_1, \ldots, v_n, w_1\}[/itex] and once again if [itex] S_1 [/itex] doesn't span V then [itex] V \backslash \text{span}(S_1)[/itex] is nonempty, so we may pick another linearly independent vector [itex] w_2[/itex] from this set. We continue this process until we find a basis.
 
  • #6
thanks everyone, i will just have to evaluate on a case-by-case basis if it is easy enough to guess a few vectors and check or if i need to find the span first like spamiam suggested.
 
  • #7
td21:

Do you know for certain that there is no method? Only way I can think of proving
such claim is some argument/proof of impossibility. Also, do you mean there are no
ways of constructing n+1 LI vectors given n LI vectors?

Method-wise, you can always use the fact that a matrix with n LI vectors will
always be similar to the nxn-identity matrix, which contains the standard basis
as columns/rows.
 

Related to How to extend a set to form a basis?

1. How do I extend a set to form a basis?

To extend a set to form a basis, you need to add linearly independent vectors until the set spans the entire vector space. This means that none of the vectors in the extended set can be written as a linear combination of the other vectors.

2. What is the difference between a set and a basis?

A set is a collection of vectors, while a basis is a specific type of set that satisfies two conditions: it is linearly independent and spans the entire vector space. In other words, a basis is the smallest set of vectors that can represent all other vectors in a given vector space.

3. How many vectors do I need to extend a set to form a basis?

The number of vectors needed to extend a set to form a basis depends on the dimension of the vector space. For a vector space of dimension n, you will need n linearly independent vectors to form a basis.

4. Can any set be extended to form a basis?

No, not every set can be extended to form a basis. The set must satisfy the two conditions of being linearly independent and spanning the entire vector space in order to be considered a basis. If a set does not satisfy these conditions, it cannot be extended to form a basis.

5. What is the significance of having a basis for a vector space?

A basis is important because it provides a unique and efficient way to represent vectors in a vector space. It also allows for easy calculations and transformations, as the coefficients of a vector in the basis can be easily determined. Additionally, having a basis allows for the study and understanding of the properties and structure of a vector space.

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