Is {(sinx)^2, (cosx)^2} a Basis of W in Linear Algebra?

In summary, the conversation discusses linear algebra and specifically the topic of basis. The first part of the conversation focuses on proving that W, a subset of F, is a vector space. This is proven by showing that all elements of W can be represented by a linear combination of three given elements. The second part of the conversation deals with finding a basis for W. The individual asking for help is having difficulty with this task, but after further discussion, it is determined that {(sinx)^2, (cosx)^2} is a basis for W and the dimension of W is 2. The conversation ends with confirmation that this solution is valid and correct.
  • #1
discoverer02
138
1
Linear Algebra -- Basis

I had the following problem on an exam this morning and it kind of threw me. I'd appreciate it if someone could review my answers and reasoning and let me know if I answered correctly.

W is a subset of F and spanned by {3, (sinx)^2, (cosx)^2}

a) Prove W is a vector space:

All w's that are members of W can be represented by: a3 + b(sinx)^2 + c(cosx)^2 = f(x)

From this obvious that f(x) is closed under addition and scalar multiplication, so I won't go into details.

b) Find a basis of W:

This is where I was having some problems.

a3 + b(sinx)^2 + c(cosx)^2 = 0;
since (sinx)^2 + (cosx)^2 = 1, there's no linear independance, but is if I calculate the Wronskian of {(sinx)^2, (cosx)^2} is show's they're linearly independent. Because {3, (sinx)^2, (cosx)^2} spanned W, but 3 is a linear combination of {(sinx)^2, (cosx)^2}, {(sinx)^2, (cosx)^2} spans W. Therefore, {(sinx)^2, (cosx)^2} is a basis of W, and the dimension of W is 2.

Is this valid and correct?

Thanks for the help.
 
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  • #2
discoverer02 said:
Is this valid and correct?
Looks good to me. Except, I'm a little uneasy how you got the Wronskian to show linear independence.
 
  • #3
Thanks for your reply.

For the Wronskian, W(x) = -Sin(2x). There's an definitely an x where W(x) is not equal to 0, so {(sinx)^2, (cosx)^2} are linearly independent.
 
  • #4
Oh, right. I wasn't paying close enough attention. For some reason I imagined a 1 in there. OK, so now it looks even better to me.
 

What is a basis in linear algebra?

A basis in linear algebra is a set of vectors that are linearly independent and can be used to represent any vector in a given vector space. It is sometimes referred to as the "building blocks" of a vector space.

How do you determine if a set of vectors is a basis?

To determine if a set of vectors is a basis, you can use the linear independence test. This involves checking if the vectors are linearly independent, meaning that none of the vectors can be written as a linear combination of the other vectors in the set.

What is the dimension of a vector space?

The dimension of a vector space is the number of vectors in a basis for that space. It is also the maximum number of linearly independent vectors that can be found in that space.

What is the difference between a basis and a spanning set?

A basis is a set of vectors that are linearly independent and can be used to represent any vector in a given vector space, while a spanning set is a set of vectors that can represent all the vectors in a vector space but may not necessarily be linearly independent.

Why is having a basis important in linear algebra?

A basis is important in linear algebra because it allows us to represent vectors in a more efficient way. It also helps us to understand and perform operations such as matrix multiplication and solving systems of linear equations.

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