- #1
victoranderson said:Please see attached question.
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)...
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.
victoranderson said:(b)
Thank you for your reply
This is my solution. I do not know if it is correct.
Assume v, ø(v) and ø(v)^2 are linearly dependent
[tex] => a_{1}v+a_{2}\phi (v)+a_{3}\phi (v)^{2} = 0 [/tex]
for [tex] a_{1}, a_{2}, a_{3} [/tex] not all zero
since [tex] \phi (v)^{2} \neq 0 [/tex] we have [tex]a_{3}=0[/tex]
Contradiction, so v, ø(v) and ø(v)^2 are linearly dependent
victoranderson said:I have another question about part c.
Why column 1 is M^2*v? How can we know?
Please see attached. Many thanks.
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##?
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
[tex] => a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v) = 0 [/tex]
for [tex] a_{1}, a_{2}, a_{3} [/tex] not all zero
[tex] => \phi^2( a_{1}v+a_{2}\phi (v)+a_{3}\phi ^{2}(v)) = 0 [/tex]
since [tex] \phi^2 \neq 0 [/tex]
it leads to contradiction
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)##??
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##
and it leads to contradiction??
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?
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
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.
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.
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.
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.
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.
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.