Two subspaces are said to be perpendicular if every vector in one subspace is orthogonal (or perpendicular) to every vector in the other subspace.
This proof helps to establish the relationship between two subspaces and their orthogonal complements. It is a fundamental concept in linear algebra and is used in various applications, such as finding solutions to systems of linear equations and understanding the properties of vector spaces.
To show that S perp contains V perp, we need to prove that every vector in V perp is also in S perp. This can be done by showing that the dot product of any vector in V perp with any vector in S is equal to 0. This implies that the vector is orthogonal to every vector in S, and therefore, is also in S perp.
Yes, in this proof, we assume that both S and V are subspaces of the same vector space. This is because the concept of perpendicular subspaces only applies when the subspaces exist within the same vector space.
Yes, there are specific properties of subspaces that can be used in this proof. One important property is that the orthogonal complement of a subspace is also a subspace. This means that if S is a subspace of V, then S perp is also a subspace of V.