An independent subspace is a subset of a vector space that contains vectors that are not linearly dependent on each other. This means that none of the vectors in the subspace can be written as a linear combination of the other vectors in the subspace.
To prove linear independence in a subspace, you can use the definition of linear independence and show that no linear combination of the vectors in the subspace can equal the zero vector, except for the trivial solution where all coefficients are equal to zero. You can also use the determinant test or the rank-nullity theorem to prove linear independence.
To disprove linear independence in a subspace means to show that at least one vector in the subspace can be written as a linear combination of the other vectors in the subspace. This would violate the definition of linear independence and prove that the subspace is not linearly independent.
No, a subspace cannot be both independent and dependent. By definition, a subspace is either linearly independent or linearly dependent. If a subspace is independent, none of the vectors in the subspace can be written as a linear combination of the other vectors. If a subspace is dependent, at least one vector can be written as a linear combination of the other vectors.
Proving or disproving linear independence in a subspace is important because it allows us to understand the structure and properties of the subspace. Linearly independent subspaces have unique properties and can be used to solve systems of equations and perform other calculations. On the other hand, linearly dependent subspaces can cause problems and inaccuracies in calculations, so it is important to identify and address them.