What Is the Dimension of the Vector Subspace Spanned by e^x and e^2x?

In summary, the conversation is about finding the dimension of a vector subspace spanned by the functions e^x and e^2x. It is necessary to specify the field the vector space is over, and in this case it is assumed to be C(R). The two functions span a space of dimension two and are linearly independent, making them suitable for solving the problem.
  • #1
Pearce_09
74
0
The functions e^x and e^2x
I have to find the dimension of the vector subspace spanned by this set.
Im not sure where to start, I do know how to solve other problems asking the same question just different function. Any help would be greatly appreciated.
thanks
a.p
 
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  • #2
Firstly this can only make sense if you specify over what field this is a vector space. Stating what the vectort space is would be good. DOesn't make the question any easier or harder but would be a good practice for you to get into.

Assume we mean C(R) the vector space of real valued functions (over R) then you have to vectors e^x and e^2x. They span a space of at most dimension two. Are they linearly independent (as funtions)? Ie are the real numbers u and v such that ue^x+ve^2x is the zero vector in the vector space. Hopefully you know what the zero vector is and see that obviously these are linearly independent vectors
 

FAQ: What Is the Dimension of the Vector Subspace Spanned by e^x and e^2x?

What is a vector subspace?

A vector subspace is a subset of a vector space that contains all the elements necessary to form a vector space. It is closed under vector addition and scalar multiplication, and contains the zero vector.

What is the dimension of a vector subspace?

The dimension of a vector subspace is the number of linearly independent vectors needed to span the subspace. In other words, it is the minimum number of vectors required to represent all the elements in the subspace.

How is the dimension of a vector subspace related to the dimension of the vector space?

The dimension of a vector subspace can never be greater than the dimension of the vector space it is a subset of. In fact, it can only be equal to or less than the dimension of the vector space. This is because a vector subspace is a smaller, contained space within the larger vector space.

Can a vector subspace have a dimension of zero?

No, a vector subspace must have at least one non-zero vector in order to be considered a subspace. Therefore, the minimum dimension of a vector subspace is one.

How can the dimension of a vector subspace be determined?

The dimension of a vector subspace can be determined by finding the maximum number of linearly independent vectors in the subspace. This can be done by using methods such as Gaussian elimination or finding a basis for the subspace.

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