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Extra info for A critique of the determination of the energy-momentum of a system from the equations of motion of matter in the general theory of relativity

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Clearly if fv1 : : : vn g is a basis for V then ImA = spanfAv1 : : : Avn g and is thus a subspace. The map A is called surjective when ImA = W . The dimension of the image space is called the rank of the linear mapping A, and the concept is applied as well to the associated matrix A]. Namely rank A] = dim(ImA): If S is a subspace of V , then the image of S under the mapping A is denoted AS . 1. Linear spaces and mappings 27 In particular, this means that AV = ImA. Another important set related to A is its kernel, or null space, de ned by ker A = fv 2 V : Av = 0g: In words, ker A is the set of vectors in V which get mapped by A to the zero element in W , and is easily veri ed to be a subspace of V .

In general each block Jk has a single eigenvalue of A in all its diagonal entries however a given eigenvalue of A may occur in several blocks. If the dimensions of Jk is nk nk , then from our earlier discussion there are exactly nk invariant subspaces of A associated with Jk . All invariant subspaces can be constructed from this collection associated with the Jordan blocks. We will not explicitly require a constructive method for transforming a matrix to Jordan form, and will use this result solely for analysis.

The dimension of Pm is more challenging to compute and its determination is an exercise at the end of the chapter. An important computational concept in vector space analysis is associating a general k dimensional vector space V with the vector space Fk . This is done by taking a basis fv1 : : : vk g for V , and associating each vector v in V with the vector of coordinates in the given basis, 2 6 4 1 3 .. 75 2 Fk : . k Equivalently, each vector vi in the basis is associated with the vector 2 3 0 6 ..

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A critique of the determination of the energy-momentum of a system from the equations of motion of matter in the general theory of relativity


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