# Show that v, ø(v), ø(v)^2 are independent

• victoranderson
I'm sorry to add this part of the code, but the last line of the proof is missing, so here it is:and since we assume v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly dependent, ##a_{1}=0##, ##a_{2}=0##, ##a_{3}=0## are not all zero.Therefore, v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly INDEPENDENT.
victoranderson
I can finish part (a)

For part b, how can I find ø(v) ?
Although I can find ø(v1) and ø(v2) but I think it is unrelated to ø(v)...

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victoranderson said:
I can finish part (a)

For part b, how can I find ø(v) ?
Although I can find ø(v1) and ø(v2) but I think it is unrelated to ø(v)...

Part b isn't really related to the first part. If you assume ##v, \phi(v), \phi^2(v)## are linearly DEPENDENT what does that tell you about them? What's the definition of linear dependence? Try to use that to reach a contradiction to what you know about the vectors. If you can show one of them must be zero then you've shown they are linearly independent.

Dick said:
Part b isn't really related to the first part. If you assume ##v, \phi(v), \phi^2(v)## are linearly DEPENDENT what does that tell you about them? What's the definition of linear dependence? Try to use that to reach a contradiction to what you know about the vectors. If you can show one of them must be zero then you've shown they are linearly independent.

(b)

This is my solution. I do not know if it is correct.
Assume v, ø(v) and ø(v)^2 are linearly dependent

$$=> a_{1}v+a_{2}\phi (v)+a_{3}\phi (v)^{2} = 0$$
for $$a_{1}, a_{2}, a_{3}$$ not all zero

since $$\phi (v)^{2} \neq 0$$ we have $$a_{3}=0$$

Contradiction, so v, ø(v) and ø(v)^2 are linearly dependent

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I have another question about part c.
Why column 1 is M^2*v? How can we know?

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• partc.jpg
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victoranderson said:
(b)

This is my solution. I do not know if it is correct.
Assume v, ø(v) and ø(v)^2 are linearly dependent

$$=> a_{1}v+a_{2}\phi (v)+a_{3}\phi (v)^{2} = 0$$
for $$a_{1}, a_{2}, a_{3}$$ not all zero

since $$\phi (v)^{2} \neq 0$$ we have $$a_{3}=0$$

Contradiction, so v, ø(v) and ø(v)^2 are linearly dependent

That's not correct at all. You did get the definition of linear dependence right. So you have ##a_{1}v+a_{2}\phi (v)+a_{3}\phi^2 (v) = 0## (notice I wrote ##\phi^2 (v)## not ##\phi (v)^2## that's what you really want). It's worth trying to take it slowly and get this right. It's pretty basic. ##\phi^2 (v) \ne 0## doesn't imply anything about ##a_3##. I'll give you a hint. What happens if you apply ##\phi^2## to that equation? What does that tell you about ##a_1##?

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victoranderson said:
I have another question about part c.
Why column 1 is M^2*v? How can we know?

It's much better to post new questions in new threads.

Dick said:
That's not correct at all. You did get the definition of linear dependence right. So you have ##a_{1}v+a_{2}\phi (v)+a_{3}\phi^2 (v) = 0## (notice I wrote ##\phi^2 (v)## not ##\phi (v)^2## that's what you really want). It's worth trying to take it slowly and get this right. It's pretty basic. ##\phi^2 (v) \ne 0## doesn't imply anything about ##a_3##. I'll give you a hint. What happens if you apply ##\phi^2## to that equation? What does that tell you about ##a_1##?

I am a beginner in this topic...I know the trick is to add phi to the equation
is this a correct proof?

Assume v, ø(v) and ø(v)^2 are linearly dependent

$$=> a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0$$
for $$a_{1}, a_{2}, a_{3}$$ not all zero

$$=> \phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0$$

since $$\phi^2 \neq 0$$

victoranderson said:
I am a beginner in this topic...I know the trick is to add phi to the equation
is this a correct proof?

Assume v, ø(v) and ø(v)^2 are linearly dependent

$$=> a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0$$
for $$a_{1}, a_{2}, a_{3}$$ not all zero

$$=> \phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0$$

since $$\phi^2 \neq 0$$

Nope, not correct. ##\phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = a_1 \phi^2(v) +a_2 \phi^3(v) + a_3 \phi^4(v)##, what can you say about ##\phi^3(v)## and ##\phi^4(v)##??

Dick said:
Nope, not correct. ##\phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = a_1 \phi^2(v) +a_2 \phi^3(v) + a_3 \phi^4(v)##, what can you say about ##\phi^3(v)## and ##\phi^4(v)##??

In my opinion, ##\phi^3(v)## = 0 => ##\phi^4(v)## = 0

so, ##a_1 \phi^2(v) +a_2 \phi^3(v) + a_3 \phi^4(v)##=0 implies ##a_1 \phi^2(v)## = 0

which gives ##a_1=0##

victoranderson said:
In my opinion, ##\phi^3(v)## = 0 => ##\phi^4(v)## = 0

so, ##a_1 \phi^2(v) +a_2 \phi^3(v) + a_3 \phi^4(v)##=0 implies ##a_1 \phi^2(v)## = 0

which gives ##a_1=0##

It's not a contradiction yet. Now put ##a_1=0## into ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0##. Any ideas what to do next?

Dick said:
It's not a contradiction yet. Now put ##a_1=0## into ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0##. Any ideas what to do next?

Thanks dick, i know the trick now.
So this is my complete proof of part (b), please have a look.

Assume v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly dependent

i.e ##a_{1}v+a_{2}\phi (v)+a_{3}\phi^2 (v) = 0## for ##a_{1}, a_{2}, a_{3}## not all zero

## \phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0 ##

=> ## a_1 \phi^2(v) +a_2 \phi^3(v) + a_3 \phi^4(v) = 0##

Since ##\phi^3(v)## = 0 => ##\phi^4(v)## = 0, we have ##a_{1}=0##

So the equation ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0 ## gives ##a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0##

## \phi (a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0 ##

=> ##a_{2}\phi ^2(v)+a_{3}\phi ^{3}(v) = 0##

and once again, since ##\phi^3(v)= 0## and ##\phi^2 \neq 0 ##

we have ##a_{2}=0 ##

so the equation ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0 ## gives ##a_{3}\phi ^{2}(v) = 0##

since ##\phi^2 \neq 0 ## we have ##a_{3}=0 ##

This leads to contradiction, since we assume ##a_{1}, a_{2}, a_{3}## not all zero

so v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly INDEPENDENT

victoranderson said:
Thanks dick, i know the trick now.
So this is my complete proof of part (b), please have a look.

Assume v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly dependent

i.e ##a_{1}v+a_{2}\phi (v)+a_{3}\phi^2 (v) = 0## for ##a_{1}, a_{2}, a_{3}## not all zero

## \phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0 ##

=> ## a_1 \phi^2(v) +a_2 \phi^3(v) + a_3 \phi^4(v) = 0##

Since ##\phi^3(v)## = 0 => ##\phi^4(v)## = 0, we have ##a_{1}=0##

So the equation ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0 ## gives ##a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0##

## \phi (a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0 ##

=> ##a_{2}\phi ^2(v)+a_{3}\phi ^{3}(v) = 0##

and once again, since ##\phi^3(v)= 0## and ##\phi^2 \neq 0 ##

we have ##a_{2}=0 ##

so the equation ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0 ## gives ##a_{3}\phi ^{2}(v) = 0##

since ##\phi^2 \neq 0 ## we have ##a_{3}=0 ##

This leads to contradiction, since we assume ##a_{1}, a_{2}, a_{3}## not all zero

so v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly INDEPENDENT

That's perfect.

1 person
victoranderson said:
so the equation ##a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0 ## gives ##a_{3}\phi ^{2}(v) = 0##

since ##\phi^2\color\red{(v)} \neq 0 ## we have ##a_{3}=0 ##

This leads to contradiction, since we assume ##a_{1}, a_{2}, a_{3}## not all zero

so v, ##\phi (v)## and ##\phi ^{2}(v)## are linearly INDEPENDENT

Dick said:
That's perfect.

Almost. Correction in red. It isn't ##\phi^2## that isn't zero, it's ##\phi^2(v)## that isn't.

## 1. What does it mean for v, ø(v), and ø(v)^2 to be independent?

When we say that v, ø(v), and ø(v)^2 are independent, it means that there is no linear relationship between them. In other words, knowing the value of one of these variables does not tell us anything about the values of the others.

## 2. How do you show that v, ø(v), and ø(v)^2 are independent?

To show that v, ø(v), and ø(v)^2 are independent, we need to prove that the correlation coefficient between any two of these variables is equal to 0. This means that there is no linear relationship between them, and they are therefore independent.

## 3. Can v, ø(v), and ø(v)^2 be dependent on each other in a non-linear way?

Yes, it is possible for v, ø(v), and ø(v)^2 to be dependent on each other in a non-linear way. This means that while there is no linear relationship between them, there may be a non-linear relationship that exists. However, to show that they are independent, we only need to prove that there is no linear relationship between them.

## 4. Why is it important to show that v, ø(v), and ø(v)^2 are independent?

It is important to show that v, ø(v), and ø(v)^2 are independent because it allows us to use each of these variables separately in our analysis without worrying about any potential confounding or overlapping effects. This helps to ensure the accuracy and validity of our scientific findings.

## 5. Are there any assumptions or requirements for showing that v, ø(v), and ø(v)^2 are independent?

Yes, there are some assumptions and requirements that need to be met in order to show that v, ø(v), and ø(v)^2 are independent. These include having a large enough sample size, ensuring that the data is normally distributed, and checking for any potential outliers or influential points. Additionally, we must also use appropriate statistical tests to confirm the independence of these variables.

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