A basis in linear algebra is a set of linearly independent vectors that span a subspace. This means that any vector in the subspace can be written as a linear combination of the basis vectors. The number of basis vectors is equal to the dimension of the subspace.
To determine if a set of vectors forms a basis, you can use the rank-nullity theorem. This theorem states that the dimension of a subspace is equal to the sum of the rank (number of linearly independent columns) and the nullity (number of free variables) of the matrix formed by the vectors. If the rank of the matrix is equal to the number of vectors and there are no free variables, then the set of vectors forms a basis.
Yes, a subspace can have multiple bases. This is because there are usually infinitely many ways to choose a set of linearly independent vectors that span a given subspace. However, all bases for a given subspace will have the same number of vectors (equal to the dimension of the subspace).
To find the coordinates of a vector with respect to a basis, you can use the change of basis formula. This involves finding the inverse of the matrix formed by the basis vectors and multiplying it by the vector. The resulting vector will have the coordinates of the original vector with respect to the basis.
Yes, it is possible to have a basis for a subspace that is not a set of orthogonal vectors. While orthogonal bases can be useful in certain applications, they are not necessary for a basis to be valid. As long as the basis vectors are linearly independent and span the subspace, they can form a basis.