An Example of a 2-Dimensional Subspace of C[0,1]

In summary, the conversation discusses finding an example of a two-dimensional subspace of continuous functions on the interval [0,1], but the speaker is unsure of how to approach the problem. They are prompted to define what "two-dimensional" means and to consider the dimension of subspace of polynomial functions.
  • #1
sheldonrocks97
Gold Member
66
2

Homework Statement



Give an example of show that no such example exists.

A two dimensional subspace of C[0,1]

Homework Equations



None that I know of.

The Attempt at a Solution



I know that C[0,1] is a set of continuous functions but I'm not sure where to go after that.
 
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  • #2
sheldonrocks97 said:

Homework Statement



Give an example of show that no such example exists.

A two dimensional subspace of C[0,1]

Homework Equations



None that I know of.

The Attempt at a Solution



I know that C[0,1] is a set of continuous functions but I'm not sure where to go after that.

Define what 'two dimensional' means. That should be a clue.
 
  • #3
Possibly a further clue: how would you find a two-dimensional subspace of ##\mathbb{R}^n##?
 
  • #4
Polynomials are continuous functions, aren't they? It should be easy to determine the dimension of a subspace of polynomial functions.
 

FAQ: An Example of a 2-Dimensional Subspace of C[0,1]

What is a 2-dimensional subspace?

A 2-dimensional subspace is a subset of a vector space that contains two linearly independent vectors. In other words, any vector in this subspace can be written as a linear combination of these two basis vectors.

How is a 2-dimensional subspace represented in C[0,1]?

In C[0,1], a 2-dimensional subspace can be represented as a set of functions that are continuous on the interval [0,1] and can be written as a linear combination of two basis functions.

What is the basis of a 2-dimensional subspace in C[0,1]?

The basis of a 2-dimensional subspace in C[0,1] can be any two linearly independent functions that span the entire subspace. These basis functions can be chosen in many ways, but commonly used ones include polynomials, trigonometric functions, and exponential functions.

How can I determine if a given set of functions form a 2-dimensional subspace in C[0,1]?

To determine if a given set of functions form a 2-dimensional subspace in C[0,1], you can check if the functions are continuous on the interval [0,1], linearly independent, and span the entire subspace. This can be done by setting up a linear system of equations and solving for the coefficients of the basis functions.

Can a 2-dimensional subspace in C[0,1] contain more than two basis functions?

Yes, a 2-dimensional subspace in C[0,1] can contain more than two basis functions. As long as these functions are linearly independent and span the entire subspace, they can form a basis for the subspace. However, using the minimum number of basis functions (in this case, two) is preferred as it simplifies calculations and makes it easier to visualize the subspace.

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