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JamesTheBond
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Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?
JamesTheBond said:Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?
JamesTheBond said:For W < V, B_W (some basis of W) is linearly independent and spans W, therefore, there exists some B_V (basis of V) s.t. B_W < B_V.
I believe this is the basis extension theorem... not sure.
mathwonk said:here is an example to think about. (1,0)n and (0,1) give a basis of R^2. now consider the subspace where y=x, i.e. the line at 45degrees through the origin. how woulkd you get a basis of that subspace from the given basis of R^2?
A subspace in a vectorspace is a subset of the original vectorspace that satisfies the properties of a vectorspace. This means that it is closed under vector addition and scalar multiplication, and contains the zero vector.
The basis of a subspace is a set of linearly independent vectors that span the subspace. This means that any vector in the subspace can be written as a linear combination of the basis vectors.
To find the basis of a subspace, first find a set of vectors that span the subspace. Then, use the Gaussian elimination method to reduce the set of vectors to a set of linearly independent vectors. These linearly independent vectors are the basis of the subspace.
Yes, a subspace can have more than one basis. This is because there can be multiple sets of linearly independent vectors that span the same subspace. However, all bases for the same subspace will have the same number of vectors, known as the dimension of the subspace.
The basis of a subspace and the dimension of the subspace are closely related. The number of vectors in the basis is always equal to the dimension of the subspace. Additionally, any set of linearly independent vectors with the same number of vectors as the dimension of the subspace can be used as a basis for that subspace.