Finding a Basis for a Subspace with a Specific Condition?

In summary, the task is to find a basis for the subspace U of V, which is generated by the functions f1, f2, and f3. The basis consists of all functions g in V that satisfy the condition g(0)=g(1). The condition also restricts the possible values of the real numbers a1, a2, and a3 in the equation g(x)=a1f1(x)+a2f2(x)+a3f3(x).
  • #1
phy
Let V be the subspace of F([0,1],R) generated by the functions f1, f2, f3 given by:

f1(x)=1/(x+1) , f2 (x) = 2-x and f3(x) = x^2

for all x element of [0,1]. Find a basis of the subspace U of V that consists of all the functions g of V such that g(0) = g(1).

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Ok now the very first question I have is what on Earth is g? We have three functions (f1, f2, and f3) so where did this g come from? Secondly, how do I start this question? I've looked in my textbook and lecture notes but there aren't any examples like this one. We only have the really simple vector subspace examples and they didn't really help much. Any suggestions would be greatly appreciated. Thanks :smile:
 
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  • #2
Does nobody know how to do this question at all?
 
  • #3
Huh? You defined what g is, so why are you asking?

Any g in U must satisfy

[tex]g(0)=g(1)[/tex]

and since U is a subset of V, g must also satisfy

[tex]g(x)=a_1f_1(x)+a_2f_2(x)+a_3f_3(x)[/tex]

where the a's are real numbers. What you will have to figure out is how the condition g(0)=g(1) restricts the possible values of a1, a2 and a3.
 

1. What is a vector subspace?

A vector subspace is a subset of a vector space that satisfies the properties of a vector space, such as closure under addition and scalar multiplication. In other words, it is a collection of vectors that can be added together and multiplied by scalars to produce other vectors within the same space.

2. How can you determine if a set of vectors is a subspace?

To determine if a set of vectors is a subspace, you need to check if it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and contains the zero vector. If it satisfies all three properties, then it is a subspace.

3. What is the difference between a vector subspace and a vector space?

A vector subspace is a subset of a vector space that satisfies the properties of a vector space, while a vector space is a set of vectors that can be added and multiplied together. In other words, a vector subspace is a smaller portion of a vector space that still behaves like a vector space.

4. Can a vector subspace contain infinite vectors?

Yes, a vector subspace can contain an infinite number of vectors. As long as it satisfies the three properties of a vector space, it can contain any number of vectors, including infinite ones.

5. How are vector subspaces used in real-world applications?

Vector subspaces are used in various fields of science, such as physics, engineering, and computer science. They are used to model and analyze complex systems, such as electric circuits, fluid dynamics, and machine learning algorithms. They also have applications in data analysis, where vectors are used to represent and manipulate data points.

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