What is the subspace spanned by a single vector in function space?

In summary, a subspace is a subset of a vector space with similar structure and operations. A vector spans a subspace if it can reach every point in the subspace through linear combinations. The span test can be used to determine if a vector spans a subspace. Multiple vectors can span the same subspace as long as they are linearly independent. Understanding this concept is important for visualizing and solving problems related to linear equations and transformations.
  • #1
cahiersujet
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What would the subspace spanned by a single vector (for example) f(x)=x+1 be?
 
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  • #2
The subspace spanned by a vector is the set of all scalar multiples of that vector. A function like f(x)=x+1 is a "vector" in some function space, most likely over the real or complex numbers, so the answer is all functions of the form g(x)=a*(x+1), where a is an element of the scalar field (R or C).
 

1. What is a subspace?

A subspace is a subset of a vector space that has the same structure and operations as the original vector space. It is closed under addition and scalar multiplication and contains the zero vector.

2. What does it mean for a vector to span a subspace?

A vector spans a subspace if all elements in that subspace can be expressed as a linear combination of that vector. In other words, the vector can reach every point in the subspace through linear combinations.

3. How do you determine if a vector spans a subspace?

To determine if a vector spans a subspace, you can use the span test. This involves checking if the vector is a linear combination of the basis vectors of the subspace. If it is, then the vector spans the subspace.

4. Can more than one vector span the same subspace?

Yes, it is possible for multiple vectors to span the same subspace. As long as all the vectors are linearly independent, they can span the subspace.

5. Why is understanding the concept of subspace spanned by a vector important?

Understanding the concept of subspace spanned by a vector is important because it allows us to visualize and manipulate vector spaces in a more efficient way. It also helps us to solve problems related to linear equations and transformations.

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