# Homework Help: Linear algebra problem check

1. Apr 3, 2013

### Mdhiggenz

1. The problem statement, all variables and given/known data

Let x and y be linearly independent vectors in R2.

If ||x||=2 and ||y||=3, what if anything can we conclude about the possible values of |xTy|

I know that ||x||*||y||cos(∅)=|xTy|

If ||x||=2 and ||y||=3, then we can get either 6 or -6, i'm not sure if this is right. It is the only logical thing I can come up with.

2. Relevant equations

3. The attempt at a solution

2. Apr 3, 2013

### Staff: Mentor

If cos(θ) were equal to 1, then ||x|| ||y|| cos(θ) would be 6, but that can't happen, given what we know about x and y. Do you see why?

Note that |xTy| is the absolute value of the dot product of x and y, so it can't be less than 0.

3. Apr 3, 2013

### Mdhiggenz

Which means if it cant be less then zero the only possible outcome is 6?

4. Apr 3, 2013

### Infrared

No. $$\|\vec{x}\cdot \vec{y}\| = 6 |\cos \theta|$$ as you have shown. What are all the possible values of $$6 |\cos \theta|$$ if $\theta$ is not zero or π?

5. Apr 3, 2013

### Mdhiggenz

Theta can be 90 pretty much only 90 or 270?

6. Apr 3, 2013

### Infrared

Why can not $\theta$ take on other values? Linearly independent does not mean orthogonal.

7. Apr 3, 2013

### Mdhiggenz

Actually using the cauchy-schwarz inequality

if x and y are vectors in either R.^2 or R.^3 then

|xTy|≤||x||*||y||

or simply

|xTy|≤6

8. Apr 3, 2013

### Infrared

Not quite. Since the two vectors are linearly independent, it can not be the case that $\cos\theta=1$ so the inequality is strict.

9. Apr 3, 2013

### Mdhiggenz

Well if cosθ=1 can not equal 1 then I see only one option. For it to be either zero or -1 but the absolute value takes care of the negative case.

10. Apr 3, 2013

### Infrared

Why can't $|cos\theta |$ equal 1/10, .38, or any other value in $[0,1)$?

11. Apr 3, 2013

### Mdhiggenz

Tbh I'm still quite unsure why cant cosθ be zero. It is the dot product that has a limitation on it. Not cosθ.

12. Apr 3, 2013

### Infrared

$cos\theta$ can be zero. Just not 1 from the condition of linear independence.

13. Apr 3, 2013

### Mdhiggenz

Ok. Well if cosθ can be zero then we know that it can in fact be 6. and cosθ can be anything less then 1?

14. Apr 3, 2013

### Infrared

Um, if $cos\theta$ is zero, then the expression is zero. $cos\theta$ can be anything between zero and one, including zero but not one. Explain why this is the case.

15. Apr 3, 2013

### Mdhiggenz

Sorry all this time I thought we were referring to theta.

If it can't be 1 because that would mean that orthogonality would no longer be there.

16. Apr 3, 2013

### Infrared

Correct. So now that you understand the constraint on the cosine, can you finish off the problem?

17. Apr 3, 2013

### Mdhiggenz

Which means the answer would be

6cos(∅)=|xTy| such that 0≤cos(theta)∠1

18. Apr 3, 2013

### Staff: Mentor

There's nothing about orthogonality in this problem. It's given that the vectors are linearly independent, which means they can't be parallel. The vectors CAN be orthogonal, but don't have to be.

19. Apr 3, 2013

### Infrared

Yes, it should have said there would no longer be linear independence. I did not catch that. Do you understand why having a cosine equal to one violates linear independence?

20. Apr 3, 2013

### Mdhiggenz

So is my above answer incorrect?

21. Apr 3, 2013

### Staff: Mentor

Some of your characters were rendering as empty square boxes. I put in what I thought you meant.
It's correct as far as you went, but a correct answer would probably entail saying something about θ rather than cos(θ).

22. Apr 3, 2013

### Mdhiggenz

Hmm well theta could be pi/2 or anything that would make it orthogonal?

23. Apr 3, 2013

### Infrared

Sure, orthogonality is allowed. What I was saying earlier was that $\theta$ could not be zero or pi, which would make the cosine one or negative one. As you know, two vectors that have zero angle between them are not independent.