Prove the following statement:

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In summary, the conversation is about determining the linear independence of three functions: sin^2x, cos^2x, and cos2x. The speaker asks for help on this problem and is reminded to follow the guidelines for posting homework questions. The speaker is also advised to clarify the notation for the functions being discussed.
  • #1
Naeem
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Q. Prove whether or not the following functions are linearly independent.

1. sin^2x ( "sine squared x")

2. cos^2x ("cosine squared x")

3. cos2x

Please any help is greatly appreciated on these parts:
 
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  • #2
Please read the guidelines for posting homework help questions.

https://www.physicsforums.com/showthread.php?t=4825

You must have had a thought on this problem already -- surely you know, say, the definition, or a relevant theorem?


P.S. it's customary to put some sort of punctuation between a trig function and its argument. You probably wanted to say cos 2x or cos(2x), not cos2x, and cos^2 x, not cos^2x.
 
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  • #3


To prove whether or not the given functions are linearly independent, we need to show that no linear combination of them can equal to the zero function, except for the trivial combination where all coefficients are zero.

First, let's consider the linear combination of these functions:

a*sin^2x + b*cos^2x + c*cos2x = 0

We can rewrite this as:

a*(1-cos^2x) + b*cos^2x + c*cos2x = 0

Next, we can use the trigonometric identity cos^2x = 1 - sin^2x to simplify the equation:

a + (b-c)*cos^2x + c*cos2x = 0

Now, we can see that in order for this equation to hold for all values of x, the coefficients must all be equal to zero. Therefore, a = b = c = 0, which is the trivial combination. This shows that the given functions are linearly independent.

To further prove this, we can also use the Wronskian determinant. The Wronskian of a set of functions is defined as:

W(f1, f2, ..., fn) = det([f1, f2, ..., fn; f1', f2', ..., fn'])

Where f1', f2', ..., fn' are the derivatives of the functions. If the Wronskian of a set of functions is non-zero, then the functions are linearly independent.

In this case, the Wronskian determinant is:

W(sin^2x, cos^2x, cos2x) = det([sin^2x, cos^2x, cos2x;2sinxcosx, -2sinxcosx, -2sin2x])

= -2sin2x(det([sin^2x, cos2x; 2sinxcosx, -2sin2x]))

= -2sin2x(-2sin^2x - 2cos^2x)

= 4sin2x(cos^2x - sin^2x)

= 4sin2x(cos2x)

Since cos2x is not equal to zero for all values of x, the Wronskian determinant is non-zero. Therefore, the given functions are linearly independent.

In conclusion, both methods of proof show that the functions sin^2x, cos^2
 

1. What does it mean to "prove" a statement?

Proving a statement means to provide evidence or logical reasoning to support the truth or validity of the statement. It involves using established principles, theories, and facts to verify that the statement is accurate.

2. How do you approach proving a statement?

The first step in proving a statement is to clearly understand the statement and its components. Then, you need to gather and analyze relevant information, such as data, experiments, or existing theories, to support the statement. Next, you should use logical reasoning to connect the evidence to the statement and show how it proves the statement to be true.

3. Is proving a statement the same as proving a hypothesis?

No, proving a statement and proving a hypothesis are different. Proving a statement involves showing the validity of a specific statement or claim, while proving a hypothesis involves testing a proposed explanation or prediction for a particular phenomenon or observation.

4. What are the key elements of a strong proof?

A strong proof should have a clear and concise statement, reliable and relevant evidence, effective use of logical reasoning, and a logical flow of ideas. It should also address potential counterarguments and provide a strong conclusion that summarizes the evidence and proves the statement.

5. Can a statement be proven 100% true?

It is difficult to prove a statement 100% true as new evidence or information may be discovered in the future that contradicts the statement. However, a strong proof can provide a high level of confidence in the truth or validity of a statement based on the available evidence and logical reasoning.

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