Spanning sets, eigenvalues, eigenvectors etc .

[phy]

spanning sets, eigenvalues, eigenvectors etc...

can anyone please explain to me what a spanning set is? I've been having some difficulty with this for a long time and my final exam is almost here.
also, what are eigenvalues and eigenvectors? i know how to calculate them but i don't understand why they are so important. thanks.

 

[matt grime]

A set of vectors S is a spanning set for a vector space V if any element of V can be written as a (finite) sum of linear combinations of S (We'll assume V's finite dimensional)
That is given v in V there exists vectors s_1,s_2...,s_r in S and elements t_1...t_r in the underlying field such that v = t_1s_1+\ldots+t_rs_r note that the order is you give me v in V, and then I pick these elements depending on the v you've given me.

As for eigenvalues, and eigenvectors, well, they encode a lot of geometrical information about the linear map. For instance if any evalue is zero the matrix is singular. If there are n distinct eigenvalues of an nxn matrix then there is a basis that diagonalizes it (if there are fewer it may or may not diagonalize, you need to know the minimal polynomial as well as the characteristic one). If we may switch emphasis, an eigenvector spans an invariant subspace. Lots of maths uses the idea of invariants. DIfferentiation is a linear map, its eigenvectors are the functions e^{kx}. Expressing things in terms of eigenvectors makes computation easier: if v= v_1+...v_m is a decomposition into eigenvectors then you can work out the image of v easily.
Then there is the fact that it might not be the eigenvalues/vectors that are the important thing but results about eigenvalues and vectors that count. If H is an Hermitian matrix it is diagonalizable, then there's Sylvester's law, the relations with determinants and traces (traces being very important in physics), generalizing these ideas leads to interesting results in operator theory and such as C* algebras that seem to be the way to think about quantum gravity and the like.

 

[d1d1]

what are the steps that need to be followed to find a spanning set for the space AX=A^TX where we are given a 5x5 matrix?

 

[disregardthat]

It is the same as the space (A-A^T)X = 0, i.e. the nullspace of A-A^T.