Linear Independence: Showing 1, sin^2(x), sin(2x) is Independent

In summary: A good way to approach this problem is to assume that there exists a non-zero solution to the equation a(1) + b(sin^2(x)) + c(sin(2x)) = 0 for some values of x, and then show that this leads to a contradiction. This would prove that the only solution is a = b = c = 0, and therefore the set is linearly independent.
  • #1
skoomafiend
33
0

Homework Statement



there is the vector space F(R) = {f | f:R -> R }
show that {1, sin^2(x), sin(2x)} is linearly independent

Homework Equations



a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.

The Attempt at a Solution



for that set to be considered linearly independent, it has to be linearly independent (a=b=c=0) for ALL values of x?

i mean, for x = 0

a(1) + b(sin^2(x)) + c(sin(2x)) = 0

0(1) + 1(0) + 1(0) = 0, and that would be a linearly dependent set since not all coefficients are 0.

but that is only one case. do i have to show that this is not valid for EVERY case? what would be a good way to approach these types of problems?

Thanks!
 
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  • #2
skoomafiend said:

Homework Statement



there is the vector space F(R) = {f | f:R -> R }
show that {1, sin^2(x), sin(2x)} is linearly independent



Homework Equations



a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.


The Attempt at a Solution



for that set to be considered linearly independent, it has to be linearly independent (a=b=c=0) for ALL values of x?
Yes. For the three functions to be linearly independent, the equation a(1) + b(sin^2(x)) + c(sin(2x)) = 0 hold for all values of x, and the only solutions for the constants must be a = b = c = 0.
skoomafiend said:
i mean, for x = 0

a(1) + b(sin^2(x)) + c(sin(2x)) = 0

0(1) + 1(0) + 1(0) = 0, and that would be a linearly dependent set since not all coefficients are 0.

but that is only one case. do i have to show that this is not valid for EVERY case? what would be a good way to approach these types of problems?

Thanks!
 
  • #3
what would be a valid way to show that the set is linearly independent?
 

1. What is linear independence?

Linear independence is a concept in linear algebra that describes a set of vectors that cannot be written as a linear combination of each other. This means that no vector in the set can be expressed as a sum of scalar multiples of the other vectors in the set.

2. How can you show that 1, sin^2(x), and sin(2x) are linearly independent?

To show that these three functions are linearly independent, we can use the definition of linear independence and show that no combination of the three functions can equal zero. This can be done by setting up a system of equations and solving for the coefficients.

3. Can you explain the geometric interpretation of linear independence?

The geometric interpretation of linear independence is that the vectors in the set are not parallel or collinear. This means that they cannot lie on the same line or be scaled versions of each other. Geometrically, linearly independent vectors span a higher-dimensional space and cannot be reduced to a lower-dimensional space.

4. Why is it important to prove linear independence?

Proving linear independence is important because it helps us understand the properties and relationships between vectors. It also allows us to solve systems of linear equations and perform other operations in linear algebra.

5. Can a set of two vectors be linearly independent?

Yes, a set of two vectors can be linearly independent if they are not scalar multiples of each other. This means that they cannot lie on the same line or be parallel to each other. However, a set of three or more vectors must be linearly independent if the first two vectors are linearly independent.

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