A basis for a subspace of R^3 is a set of vectors that can be used to span the entire subspace. This means that every vector in the subspace can be written as a linear combination of the basis vectors.
To find a basis for a subspace of R^3, you can use the Gaussian elimination method to reduce the subspace to its simplest form. The nonzero rows of the reduced form will form the basis for the subspace.
The dimension of a subspace of R^3 is the number of basis vectors needed to span the subspace. This can also be thought of as the number of linearly independent vectors in the subspace.
Yes, a subspace of R^3 can have multiple bases. This is because there are infinitely many ways to choose a set of vectors that can span a subspace. However, all bases for a given subspace will have the same number of basis vectors.
Finding a basis for a subspace of R^3 is important because it allows us to represent all vectors in the subspace in a concise and efficient way. It also helps with solving linear systems and understanding the properties of the subspace.