Showing linear independence, correct logic?

In summary, the conversation discusses a proof that {u, Bv} is a basis of R^2 and that R^2 is a direct sum of the subspaces generated by U = <u> and V = <v>. The conversation also addresses the issue of assuming B is not zero and the implications for r1 and r2 in the equation r1*u + r2*Bv = 0.
  • #1
cookiesyum
78
0

Homework Statement



Let u and v be two nonzero vectors in R^2. If there is no c E R such that u = cv, show that {u, Bv} is a basis of R^2 and that R^2 is a direct sum of the subspaces generated by U = <u> and V = <v> respectively.

Homework Equations



Clearly, u and v are linearly independent. So...

The Attempt at a Solution



I want to show r1u + r2Bv = 0 implies r1 = r2 = 0. If B=0 (since v cannot equal 0), then the second term drops out and r1u =0. Because u is non-zero, r1=0. Back to the original statement, if r1=0, then the first term drops out and r2Bv = 0. Since v is non-zero, and assume B is non zero as well, r2 = 0. But what if B IS zero? I'm getting a little confused by this logic...
 
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  • #2
If B=0 then {u,Bv}={u,0} is NOT a basis. So, yes, you can assume B is not zero. So assume r1*u+r2*B*v=0. If r1=0 then r2*B*v=0. What can you conclude about r2? If r1 is NOT equal to zero then u=-r2*B*v/r1. What's wrong with that?
 
  • #3
Then u is no longer non-zero! It makes sense, now. Thanks very much!
 
  • #4
cookiesyum said:
Then u is no longer non-zero! It makes sense, now. Thanks very much!

No, you aren't quite getting it. The problem with writing u=-r2*B*v/r1 is that you were given that there is no number c such that u=c*v.
 
  • #5
Oh ok, I see it. I should have read the question more carefully. Thanks for all the help!
 

Related to Showing linear independence, correct logic?

1. What does it mean for a set of vectors to be linearly independent?

Linear independence refers to the property of a set of vectors where none of the vectors can be written as a linear combination of the others. In other words, no vector in the set can be expressed as a combination of the other vectors using scalar multiplication and addition.

2. How do you show that a set of vectors is linearly independent?

To show that a set of vectors is linearly independent, you can use the definition of linear independence and perform a proof by contradiction. Assume that the vectors can be written as a linear combination of each other and then prove that this assumption leads to a contradiction.

3. Can a set of two vectors be linearly independent?

Yes, a set of two vectors can be linearly independent if they are not scalar multiples of each other. This means that they cannot be written as a linear combination of each other.

4. Why is it important to show linear independence?

Showing linear independence is important because it helps us understand the properties of a set of vectors. It also allows us to determine if a set of vectors can span a particular space, which is useful in many areas of mathematics and science.

5. Is linear independence the same as orthogonality?

No, linear independence and orthogonality are not the same. Linear independence refers to the property of a set of vectors, while orthogonality refers to the property of two vectors being perpendicular to each other. However, a set of orthogonal vectors is always linearly independent.

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