Vector subspace and eigenspace

A vector subspace is a subset of a vector space that is itself a vector space.

More formally, the vector subspace  of the vector space satisfies the following conditions:

1) Commutative and associative addition for all elements of the closed set.



2) Associativity and distributivity of scalar multiplication for all elements of the closed set




where c_1 and c_2 are scalars.

3) Scalar multiplication identity.


4) Additive inverse.


5) Existence of null vector , such that


6) Closed under addition: the sum of any two or more vectors in is another vector in .

7) Closed under scalar multiplication: the product of any vector in with a scalar is another vector in .

For example, in the  space, a plane through the origin is a subspace, as is a line through the origin. Hence, and are subspaces of . The entire space and the single point at the origin are also subspaces of the space. This implies that a subspace contains a set of orthonormal basis vectors.

An eigenspace is the set of all eigenvectors associated with a particular eigenvalue. In other words, it is a vector subspace formed by eigenvectors corresponding to the same eigenvalue. Consider the eigenvalue equation , where . If the eigenvalues of ,  and are , and respectively, then  and are the eigenspaces of the operator . Since an eigenspace is a vector subspace, it must contain a set of orthonormal basis vectors.



Show that all orthonormal basis eigenvectors in an eigenspace are linearly independent of one another.


A set of eigenvectors is linearly independent if the only solution to eq1 is when for all . Taking the dot product of eq1 with gives

which reduces to because . Since is arbitrary, we conclude that and that the set is linearly independent.


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