How to Extend and Calculate a Basis for the Whole Space

In summary, the conversation discusses finding a basis for R^3 when given two vectors in a plane by taking the cross product. The conversation then moves on to discussing how to find a basis for R^n when given n-1 vectors, with the suggestion to use a change of coordinates to make the given vectors each have a 1 in the n-th position. The conversation ends with a question about how to do this change of basis.
  • #1
boderam
24
0
For example, say you start out with (2,1,0) and (2,0,2). Well the easiest answer here is to think of these two vectors in a plane, so you should take the cross product to get the vector that is not in the plane, and there you have a basis for R^3. But how about when we run into similar problems in R^n, not just when we are given n-1 vectors, but perhaps any m less than n. What would be the systematic method?
 
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  • #2
Do a change of coordinates so that the given n vectors are each zero except for a 1 in the n-th position. It should then be obvious how to extend.
 
  • #3
How do you do the change of basis so this happens? My memory is vague about this.
 

FAQ: How to Extend and Calculate a Basis for the Whole Space

How do you extend a basis for the whole space?

The process of extending a basis for the whole space involves adding linearly independent vectors to an existing basis until it spans the entire vector space. This can be done by finding a new vector that is linearly independent from the current basis and repeating the process until the basis spans the entire space.

What is the importance of extending a basis for the whole space?

Extending a basis for the whole space is important because it allows us to represent any vector in the vector space using a linear combination of the basis vectors. This is useful in many applications, such as solving systems of linear equations and performing transformations.

How can you determine if a basis spans the entire space?

A basis spans the entire space if every vector in the space can be written as a linear combination of the basis vectors. This means that the vectors in the basis must be linearly independent and the number of basis vectors must be equal to the dimension of the vector space.

Can a basis for the whole space be calculated for any vector space?

Yes, a basis for the whole space can be calculated for any finite-dimensional vector space. This is because any finite-dimensional vector space has a finite number of linearly independent vectors, which can be used to form a basis.

Is there a unique way to extend and calculate a basis for the whole space?

No, there are multiple ways to extend and calculate a basis for the whole space. The process of extending a basis is not unique and can depend on the choice of vectors and the order in which they are added. However, the resulting basis will always span the entire space as long as the initial vectors are linearly independent.

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