# 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.

## 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...

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?

Then u is no longer non-zero! It makes sense, now. Thanks very much!

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.

Oh ok, I see it. I should have read the question more carefully. Thanks for all the help!

## 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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