- #1
mathmathmad
- 50
- 0
Homework Statement
How to determine if a set of vectors span a space in general?
say, V=R^n and you're given a few vectors and asked to determine if they span the space..
how do you do that?
mathmathmad said:Homework Statement
How to determine if a set of vectors span a space in general?
say, V=R^n and you're given a few vectors and asked to determine if they span the space..
how do you do that?
When a set of vectors spans a space, it means that every vector within that space can be written as a linear combination of the given set of vectors. In other words, the set of vectors contains enough information to represent every possible vector within that space.
The process for determining if a set of vectors spans a space involves creating a system of equations where the coefficients of the linear combination are unknown. If this system of equations has a solution for every vector in the space, then the set of vectors spans the space.
Yes, a set of vectors can span more than one space. This is because the same set of vectors can be used to represent different combinations of vectors in different spaces. However, the set of vectors must be able to represent every vector within each space in order to span both spaces.
If a set of vectors does not span a space, it means that there are some vectors within that space that cannot be written as a linear combination of the given set of vectors. This can happen if the set of vectors is not large enough or if they are not diverse enough to represent all possible vectors in the space.
Yes, a set of vectors can span an infinite-dimensional space. This means that the set of vectors has enough information to represent every possible vector within that space, even if there are an infinite number of dimensions. This is often the case with vector spaces in higher-level math and physics.