I've always particularly liked the bra-ket notation for making this sort of problem intuitive, typically used by quantum physicists. (There's a wikipedia articl

I've always particularly liked the bra-ket notation for making this sort of problem intuitive, typically used by quantum physicists. (There's a wikipedia article on this notation, though I don't think it does it justice.) In that notation, a vector |a> which is basis independent has components <i|a> in some basis |i> , where (say) |i_1> , |i_2> are the basis vectors. Then to change to another basis <k| the "operator one" (which is the outer product of a basis, i.e. sum over i of |i><i|) is inserted to get <k|a> = sum over i <k|i><i|a> , which turns into the matrix mechanics. (Each of the outer products e.g. |1><1| is a "projection operator" which projects a vector onto that basis vector. The sum of all of them projects onto the whole space spanned by the vectors, which is the same as doing nothing, which is therefore the identity operator.) Once you get your head around the connections between coordinates (i.e. a_x = a_1) and the dot product with a basis vector (i.e. dot(i_1, a) = <i_1 | a> = <1|a>), this notation can make the whole thing intuitive and mechanical. I have an explanation of this online at http://cs.marlboro.edu/talks/bra_ket.pdf http://cs.marlboro.edu/talks/bra_ket.pdf .