Prove any three vectors of R^3 are linear independent

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Discussion Overview

The discussion centers around the linear independence of vectors in R^3 and the implications for spanning the space. Participants explore the relationship between linear independence, spanning, and the concept of a basis in vector spaces.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that a set of three vectors in R^3 can be linearly independent and span the space, while others argue that linear independence does not necessarily imply spanning.
  • One participant asserts that since it is not possible to have more than three linearly independent vectors in R^3, any set of three vectors must span R^3 and form a basis.
  • Another participant clarifies that a basis must satisfy the conditions of spanning and linear independence, and if either condition is met, the third follows.
  • Participants discuss the span of specific vectors in R^2 and the process of expressing any vector in terms of a linear combination of those vectors.
  • There is a question about the derivation of a specific equation related to the span of two vectors, indicating some uncertainty in understanding the notation and its implications.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between linear independence and spanning in R^3. While some assert that three linearly independent vectors must span the space, others maintain that linear independence and spanning are distinct properties that do not always coincide.

Contextual Notes

There are unresolved assumptions regarding the definitions of linear independence and spanning, as well as the implications of these concepts in different dimensions. The discussion also reflects varying levels of understanding among participants regarding vector equations and their solutions.

Who May Find This Useful

This discussion may be useful for students studying linear algebra, particularly those grappling with the concepts of vector independence, spanning sets, and bases in vector spaces.

philipc
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I was just wondering if I can make this statement, if I was to prove any three vectors of R^3 are linear independent, can I also say those three vectors span R^3?

Philip
 
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No; a set of vectors can be linearly independent without spanning a space, and a set of vectors can span a space without being linearly independent. A set of vectors that is both linearly independent and spans a space is a basis for that space.

- Warren
 
Yes Warren, but it isn't possible to have more than three linearly independent vectors in R3, therefore any set of three will span and be a basis.

I think you just lost the forest for the trees :wink:
 
Ok thanks for your help, can I ask one more, well maybe only one :biggrin:

I'm looking at my notes and kind of stuck with something,

Lets say I have the vectors [1,0] and [-1,1] and I'm asked what is the span?
so I have a[1,0] +b[-1,0] = [a-b,b] then it says this spans R^2 because [x,y] = [(x+y)-y,y] so x+y=a and y=b

not sure if I was sleeping during class and wrote the notes down wrong or what, but I'm not sure where the [x,y] = [(x+y)-y,y] came from?
Thanks again for your help
Philip
 
You've shown how to write any vector in R^2 as a combination of [1,0] and [-1,1], therefore is spans. [x,y] = (x+y)[1,0] + y[-1,1]
 
but it was not the work of me?

so is that saying I need to find values for a and b to make it = [x,y], and that can be any combo of x and y for the values of a and b? Sorry to sound so dumb, but not really sure if I'm getting it.
Philip
 
yep, you need to solve the vector equations a[1,0]+b[-1,1] = [x,y]

or splitting it into components

a-b=x
b=y

ie b=y, and a-y=x, a=x+y
 
OK now I see where it came from, thanks for the help.
Philip
 
Any basis for a vector space has three properties:

1. It spans the space
2. The vectors in it are linearly independent
3. The number of vectors in it is the same as the dimension of the space.

If any two of those are true, then the third is also true.

If you know that a set of three vectors in R3 is independent, then the must span the space.

(Strictly speaking, we define a basis as a set of vectors satisfying (1) and (2), show that any basis must have the same number of vectors and then define the dimension of the space to be that number.)
 
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