- #1
atakel
- 7
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Let t€R be fixed. Show that {u,v}CR^2 with u=(cost,sint), v=(-sint,cost) is a linearly inpedendent set.
Proving Linear Independence: Fixed t€R with {u,v}CR^2
- Thread starter atakel
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- Independent Linearly
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Homework Help Overview
The discussion revolves around proving the linear independence of the set {u,v} in R², where u=(cos t, sin t) and v=(-sin t, cos t) for a fixed t in R. Participants are exploring the concepts related to linear algebra, particularly focusing on the properties of vectors and matrices.
Discussion Character
- Exploratory, Conceptual clarification, Mathematical reasoning
Approaches and Questions Raised
- Participants discuss setting up the vectors as a 2x2 matrix and question how to determine linear independence from that matrix. There are inquiries about the concept of "rank" and its relevance to the problem. Some participants express uncertainty about their understanding of linear algebra concepts.
Discussion Status
The conversation is ongoing, with some participants offering guidance on how to approach the problem, while others express their lack of foundational knowledge in linear algebra. There is a recognition of the need for a better understanding of the subject before progressing further with the questions.
Contextual Notes
One participant mentions a lack of information about the "rank" of a matrix in the original question, indicating potential gaps in the problem setup. Additionally, there are constraints noted regarding the participant's background in linear algebra and their timeline for learning the material.
- #2
Pen_to_Paper
- 11
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My advice to you is to set up vectors u,v as a 2x2 matrix and plug in values for t;
then how do you conclude that a matrix has linearly independent vectors?
then how do you conclude that a matrix has linearly independent vectors?
- #3
atakel
- 7
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ı don't know ://
- #4
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Do you know what "the rank" of a matrix is?
- #5
atakel
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I uploadded the original question, there is no information about 'the rank'. In addition that I don't know how to solve the others.
- #6
- 22,170
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So you've never heard of the rank of a matrix? Hmm, that makes it a bit more difficult.
Anyway, our u and v are independent iff
\alpha u+\beta v=0~\Rightarrow~\alpha=\beta=0
were alpha and beta are real number.
So, how do you proceed. You assume that there exists alpha and beta such that
\alpha (\cos t,\sin t)+\beta (-\sin t,\cos t)=(0,0)
this will give you a system of two equations and two unknowns (the alpha and beta).
Solve this system. If you find \alpha=\beta=0, then our u and v are independent.
Anyway, our u and v are independent iff
\alpha u+\beta v=0~\Rightarrow~\alpha=\beta=0
were alpha and beta are real number.
So, how do you proceed. You assume that there exists alpha and beta such that
\alpha (\cos t,\sin t)+\beta (-\sin t,\cos t)=(0,0)
this will give you a system of two equations and two unknowns (the alpha and beta).
Solve this system. If you find \alpha=\beta=0, then our u and v are independent.
- #7
atakel
- 7
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thank u=)
if u have any idea about other questions, can u help me?
I didn't take a linear cource.. Solving these questions is really diffucult for me://
if u have any idea about other questions, can u help me?
I didn't take a linear cource.. Solving these questions is really diffucult for me://
- #8
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Yes, Id be happy to help you with these questions. But first you need to show me what you did yourself to try and solve these questions...
- #9
atakel
- 7
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Actually, at first I started to study ''Partial Differential Equations in Action, Salsa'' but my backround is not enough to understand all subjects. Now I turned back and I started studying linear algebra..
For now, I don't have any idea about solving these questions. I have one week to learn all these subjects..
For now, I don't have any idea about solving these questions. I have one week to learn all these subjects..
- #10
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I'm sorry to say this, but you should first learn linear algebra and then start worrying abou th questions. We cannot help you until you learned some linear algebra 
- #11
atakel
- 7
- 0
ok thank u:(
- #12
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But hey, don't despair. If you're stuck on something, you can always ask us 
- #13
atakel
- 7
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=) I am going to finish all subjects asp.. anf then I will turn back with my new questions.. thank u very much=)
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