Here's one example that can be proved in the format of Condition_1 => Condition_2 => ... => Condition_27 => Condition_1 : Theorem ------- Let A be an n×n matrix, and let T : R^n → R^n be the linear transformation defined by T(x) = Ax for x ∈ R^n. Then the following are equivalent: • A is invertible. • There exists an n × n matrix B such that BA = I_n. • There exists an n × n matrix C such that AC = I_n. • Every column of A is a pivot column. • Every column of rref(A) is a pivot column. • rref(A) has no zero rows. • rref(A)=I_n. • The only solution to Ax=0 is x=0. • For every b ∈ R^n, Ax = b has at least one solution x ∈ Rn. • For every b ∈ R^n, Ax=b has at most one solution x∈R^n. • rank A=n. • nullity A = 0. • Null(A) is the zero subspace of R^n. • The columns of A span R^n. • The columns of A are linearly independent. • The columns of A are a basis for R^n. • T is onto. • T is one-to-one. • T is invertible. • im(T) =R^n. • rank T = n. • nullity T = 0. • ker(T) is the zero subspace of R^n. • A is a product of elementary matrices. • det(A) is not equal to 0. • 0 is not an eigenvalue of A. • 0 is not an eigenvalue of T. --Dan
On Mar 23, 2015, at 10:56 AM, Dan Asimov <asimov@msri.org> wrote:
There are a lot of theorem of the form
The following are equivalent: Condition_1, ..., Condition_N.
Occasionally the standard proof of such a theorem is of the form
Condition_1 => Condition_2 => ... => Condition_N => Condition_1 .
--Dan
On Mar 23, 2015, at 2:29 AM, Guy Haworth <g.haworth@reading.ac.uk> wrote:
Let us say that, if Theorem 1 (T1) can be used to prove Theorem 2 (T2), then T1 --> T2.
Are there simple examples of 'equivalence', T1 --> T2 and T2 --> T1 (setting up loop T1 --> T2 --> T1) that could be explained to school children?
Are there cases of longer loops T1 --> ... --> Tn --> T1 which cannot, or cannot naturally, be shortened ? Schoolchildren-compatable preferred again !
Thanks - Guy
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