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We can write out the scalar matrix as such:Only if all the lambda values, when multiplied by the energy matrix, scale each energy value proportionally will we have a vector space. I will show that this is only the case for escape and orbital energy as would be expected. I will also show why relativistic mass is an issue and the problems it brings with it.EDIT: The above assumes we are considering lambda_1 = lambda_2 = lambda_3. Which will not be the case for all 3 energy values.

I guess you have no idea about linear algebra either.

: WikiVector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spacesand Hilbert spaces, which are fundamental in mathematical analysis.

: WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.

I can't answer your OP question Jeff, but I am curious as to this concept of vector spaces and can at least stay 'near' topic...Quote : WikiVector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spacesand Hilbert spaces, which are fundamental in mathematical analysis....andQuote : WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces.Could g (gravity) and a (acceleration) be described as force vectors of the same type?Or is a (acceleration) a 'real multiplier'?

: WikiEuclidean vectors are an example of a vector space. They represent physicalquantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector