Tuesday, June 5, 2012

Declaration of linear independence

I've been very busy (pinyin for beginners: Wo hen mang, simplified: 我很忙) so let's have some nerdy fun. Via mathbabe's blog, the declaration of linear independence:


When, in the course of a proof, it becomes necessary for a set to dissolve the argument which has connected it with a theorem, and to assume among the powers of mathematics a position above that of the mathematician, a decent respect for the axioms requires that a rigorous justification be given.
We hold these truths to be self-evident: that all nonzero vectors are created equal; that they are endowed by their definer with certain unalienable rights; that among these are the laws of logic and the pursuit of valid proofs; that to secure these rights, logical arguments are created, deriving their just powers from axioms; that whenever any argument becomes destructive of these ends, it is the right of the vectors to alter or to abolish it, and to institute a new argument, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to reach the correct conclusion. Prudence, indeed, will dictate that theorems long established should not be changed for light and transient causes, and accordingly all experience hath shown that sets are more disposed to accept the conclusions of arguments than to right themselves by abolishing the arguments. But when a long train of abuses and usurpations, pursuing invariably the same object, evinces a design to reduce them to zero in a non-trivial way, it is their right, it is their duty to throw off such argument, and to provide new proofs for their future security. Such has been the patient sufferance of these vectors, and such is now the necessity which constrains them to alter these arguments. The history of Professor Eigen is a history of repeated injuries and usurpations, all having in direct object the establishment of dependence among these vectors. To prove this, let facts be submitted to a candid world.
He has refused to acknowledge that he obtained a zero matrix only by multiplying our coordinate matrix by a zero matrix.
He has restricted our freedom of movement by requiring us all to live in the same hyperplane, even though we cannot all fit in one.
He has attempted unsuccessfully to invert our coordinate matrix, and, having overlooked the inverse, has concluded that the coordinate matrix is singular.
He has changed bases repeatedly for opposing with manly firmness his attempts to place us in the span of fewer vectors than the dimension of the space.
He has erected a multitude of new formulas and sent hither swarms of new functions to force our directions into a proper subspace of the vector space.
He has kept among us vectors to be orthogonal to all of us without the consent of those of us whose dot product with them is nonzero.
He has abdicated the axioms here by committing mathematical errors in computing a zero determinant for our coordinate matrix.
In every stage of these oppressions we have petitioned for redress in the most humble terms; our repeated petitions have been answered only by repeated injuries.
A mathematician whose arguments are thus marked by every error is unfit to prove the theorem which he attempts to prove.
Nor have we been wanting in attentions to Professor Eigen. We have warned him from time to time of flaws in his arguments. We have reminded him of the circumstances of our definition, we have appealed to his knowledge of the axioms, and we have requested him to disavow these usurpations which would inevitably destroy the validity of his arguments. He has been deaf to the voice of logic. We must therefore acquiesce in the necessity which denounces our separation and hold him as we hold the rest of mathematicians, an enemy when he is wrong, a friend when he is right.
We, therefore, the members of set S in vector space V, appealing to the supreme judge of mathematics for the rectitude of our intentions, do solemnly publish and declare that these vectors are, and of every right ought to be, a free and independent basis; that they are absolved from all subjection to Professor Eigen's theorems, and that all restriction of them to a hyperplane is, and of right ought to be, totally dissolved; and that as a basis, they have full power to span the space, form invertible coordinate matrices, give unique linear combinations equal to a given vector, and to do all other acts and things which a basis may of right do.
And for the support of this declaration, with a firm reliance on the protection of the properties of a vector space, we mutually pledge to each other our magnitudes, our directions, and our sacred honor.
In witness whereof we have signed our coordinates with respect to an appropriate orthonormal basis, and found them to constitute a triangular matrix with nonzero diagonal elements.

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