What is the Dimension and Linear Independence of Subspaces?

In summary, the conversation discusses the concept of bases outside of R^n and the difficulty in understanding them. The speaker mentions having a handle on bases in R^n, but struggles with understanding them in other contexts. They provide examples involving finding the dimension of subspaces spanned by given functions, but express uncertainty in how to approach these problems. The other person suggests starting with the definitions of span and linearly independent, and the speaker expresses gratitude for the input.
  • #1
seang
184
0
I'm having trouble understanding bases outside of R^n; I've got a handle on those I think.

Here are some examples:

13. In C(-Pi,Pi), find the dimension of the subspace spanned by 1, cos(2x), and cos^2(x).

14. Find the dimension of the subspace of P3 spanned by x, x-1 , x^2+1, x^2-1

I really don't know how to begin either of these

Thanks for any input.
 
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  • #2
Begin with the definition of a span: http://mathworld.wolfram.com/VectorSpaceSpan.html" .
 
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  • #3
hmm I already know about spans but that might have helped actually. thanks.
 
  • #4
More important, I think, is the definition of "linearly independent". How many of those given functions are linearly independent?
 

What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the space can be written as a linear combination of the basis vectors.

How do you find the dimension of a vector space?

The dimension of a vector space is the number of vectors in its basis. It can be found by counting the number of linearly independent vectors in the space, or by finding the maximum number of linearly independent columns in the matrix that represents the space.

What is the difference between basis and dimension?

Basis refers to a specific set of vectors that span a vector space, while dimension refers to the number of vectors in the basis. In other words, basis is a set and dimension is a number.

Can a vector space have more than one basis?

Yes, a vector space can have infinitely many bases. However, all bases for a given vector space will have the same number of vectors, which is the dimension of the space.

How can you determine if a set of vectors is a basis for a vector space?

To determine if a set of vectors is a basis for a vector space, you can check if the vectors are linearly independent and if they span the entire space. If both conditions are met, then the set of vectors is a basis for the space.

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