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Xingconan
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Homework Statement
The Attempt at a Solution
I don't really know how to do this, so I hope someone can give some hints or briefly tell me what I should do.
A basis for a subspace is a set of linearly independent vectors that span the subspace. This means that all vectors in the subspace can be written as a linear combination of the basis vectors.
To find a basis for a subspace, you can use the row reduction method to reduce the subspace's matrix representation to its reduced row echelon form. The nonzero rows in this form will be the basis vectors for the subspace.
The dimension of a subspace is the number of vectors in a basis for that subspace. In other words, it is the minimum number of vectors needed to span the subspace.
Yes, a subspace can have an infinite number of bases. This is because there can be multiple sets of linearly independent vectors that span the same subspace.
Finding a basis for a subspace is important because it helps us understand the structure and properties of the subspace. It also allows us to easily perform calculations and operations on vectors within that subspace.