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I was wondering what the difference between a generalized eigenspace for an eigenvalue and just an eigenspace is. I know that you can get a vector space using an eigenbasis ie using the eigenvectors to span the space but apart from that im kinda stumped.

Also with regard to this i was trying to answer the question: Show that if U is the generalised eigenspace for an eigenvalue a and V is the generalised eigenspace for an eigenvalue b then if a doesn't equal b, U intersects V only in the zero vector. Now i understand the basic premise, if you have two different eigenvalues you need to show that their eigenvectors are linearly independent and thus can be used to span two non overlapping spaces (i know overlapping is more for venn diagrams but thats how i think about many of these problems). What I dont understand is this: if we have an operator A on Rn the whole space is the direct sum of the generalised eigenspaces. I guess my question here is more about direct sums actually. If we 'add' two spaces together, we're not actually adding them are we? Instead we're constructing a new basis which is the union of the two basis sets of the two different spaces and building a new space from that. The reason i ask is if we have two LI vectors in R2, the union of those spaces would just be two lines rather than the whole space (yes i understand the concept of spanning spaces and stuff) so im assuming when we say direct sum we mean, effectively, the space spanned by those two vectors.

Does this sort of make sense?

Thanks

-Graeme

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# Eigenspaces, eigenvalues and eigenbasis

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