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anonymous
 5 years ago
example finding the basis and dimension of the subspace of X of a 2 x 2 matrix?
anonymous
 5 years ago
example finding the basis and dimension of the subspace of X of a 2 x 2 matrix?

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0What exactly do you need? Do you have a specific question?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0If you post your question and I'm not around, I'll get an email about it and I'll help you out as soon as I can.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0The question is  finding the basis and dimension of the subspace of X of a 2 x 2 matrix?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0The basis will be dependent upon the form of the subspace. I mean, if we're taking a normal 2 x 2 matrix, M = a b c d the standard basis is the set: B = { 1 0, 0 1, 0 0, 0 0} 0 0 0 0 1 0 0 1 and the dimension is, by definition, the cardinality of the basis (i.e. the number of elements in the basis). So \[\dim(M_{2 \times 2})=4\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0You can get basis sets that have a different number of elements (and therefore, dimension) depending on special conditions. For example, if the 2 x 2 matrix is to be symmetric, then you'd have, S = a b = a1 0 + b0 1 + c0 0 b c 0 0 1 0 0 1. Therefore the set { 1 0 , 0 1 , 0 0 } 0 0 1 0 0 1 spans the subset of all symmetric 2 x 2 matrices, and can be shown to be linearly independent, and so forms a basis for this subset. The dimension in this case would be 3.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Incidentally, it's easy to show that these basis elements are linearly independent, since by the definition of linear independence, a set of vectors in linearly independent if the equation \[c_1v_1+c_2v_2+...+c_nv_n=0\]has only the trivial solution,\[c_1=c_2=...=c_n=0\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0For the first basis, letting each of the v_i be one of the matrices, \[c_1v_1+c_2v_2+c_3v_3+c_4v_4\] = c_1 c_2 c_3 c_4 = 0 0 0 0 if and only if\[c_1=c_2=c_3=c_4=0\]the trivial solution.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0You can do similar with the second basis.
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