Proving Linear Independence and Span of B={(1,-1,-1),(1,0,1),(0,-1,1)} in R^3

In summary, the problem asks to show that the set B={(1,-1,-1),(1,0,1),(0,-1,1)} is a basis for the vector space R^3, meaning that it is both linearly independent and spans R^3. This can be proven using the concept of dimension or by directly showing that the three equations have a solution for any real numbers x, y, and z.
  • #1
DanielFaraday
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Homework Statement



Prove that B={(1,-1,-1),(1,0,1),(0,-1,1)} spans R^3.

(Actually, the problem asks to show that B is a basis for R^3. This would require that I prove linear independence AND that it spans R^3. The first is easy, but I'm not sure about the second.)

Homework Equations



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The Attempt at a Solution



I know how to prove that these vectors are linearly independent, but does that prove that B spans real space?
 
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  • #2
It has to do with the notion of "dimension" ... If R^3 has dimension 3, and you have a linearly independent set of 3 vectors, then that set spans R^3.
 
  • #3
Oh, I see. So the fact that they are 3 linearly independent vectors means that they do span R^3. Got it. Thanks!
 
  • #4
If you have that theorem:
"A basis for vector space has three properties:
1) It spans the space
2) It is a set of independent vectors
3) The number of vectors in the basis is equal to the dimension of the vector space"
and you know that R3 has dimension 3, then you can say "this is a set of 3 independent vectors so it spans R3."

If you do not have all of those facts, you could do it directly:
Show how to find, for any real numbers, x, y, z, real numbers, a, b, c, such that
a(1,-1,-1)+ b(1,0,1)+ c(0, -1, 1)= (x,y,z). That is the same as showing that the three equations, a+ b= x, -a- c= y, -a+ b+ c= z, has a solution (not necessarily unique) no matter what x, y, and z are.
 

1. What is the definition of a "span" in linear algebra?

A span in linear algebra refers to the set of all possible linear combinations of a given set of vectors. In other words, it is the set of all possible vectors that can be created by multiplying each vector in the set by a scalar and adding them together.

2. How is the span of a vector or set of vectors related to linear independence?

The span of a vector or set of vectors is directly related to linear independence. If the span of a set of vectors is equal to the number of vectors in the set, then the set is considered linearly independent. In other words, none of the vectors in the set can be written as a linear combination of the other vectors in the set.

3. Can a single vector span a space in linear algebra?

No, a single vector cannot span a space in linear algebra. A span requires at least two vectors to create a linear combination and produce a new vector that is not already in the set. Therefore, a single vector can only span a line in two-dimensional space or a plane in three-dimensional space.

4. How can the span of a set of vectors be visualized?

The span of a set of vectors can be visualized as a subspace in the corresponding dimensional space. For example, in two-dimensional space, the span of two linearly independent vectors would create a plane, while the span of three linearly independent vectors would create the entire two-dimensional space.

5. What is the significance of the span in linear algebra?

The span plays a crucial role in understanding the linear independence of a set of vectors and in solving systems of linear equations. It helps determine the dimension of a vector space and is essential in many applications of linear algebra, such as machine learning and computer graphics.

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