M.4. Finitely generated Modules over a PID, part I
... of the first type gives a matrix with r in the (i, 1) position. Then interchanging the first and i–th rows yields a matrix with r in the (1, 1) position. Since d(r) < d(α), we are done. Proof of Proposition M.4.7. If A is the zero matrix, there is nothing to do. Otherwise, we proceed as follows: Ste ...
... of the first type gives a matrix with r in the (i, 1) position. Then interchanging the first and i–th rows yields a matrix with r in the (1, 1) position. Since d(r) < d(α), we are done. Proof of Proposition M.4.7. If A is the zero matrix, there is nothing to do. Otherwise, we proceed as follows: Ste ...
(pdf)
... Proof. Given Proposition 2.9 it suffices to show that Mn (R) ⊂ gln (R). In order to prove that Mn (R) is a subset of gln (R) works, we want to show that for every A in Mn (R), A also belongs to Mn (R), which means that there exists a path γ : (−, ) → GLn (R) such that γ(0) = I and γ 0 (0) = A. No ...
... Proof. Given Proposition 2.9 it suffices to show that Mn (R) ⊂ gln (R). In order to prove that Mn (R) is a subset of gln (R) works, we want to show that for every A in Mn (R), A also belongs to Mn (R), which means that there exists a path γ : (−, ) → GLn (R) such that γ(0) = I and γ 0 (0) = A. No ...
Textbook
... 1. For a general matrix Am×n describe what the following products will provide. Also give the size of the result (i.e. "n × 1 vector" or "scalar"). a. Ae j b. eiT A c. eiT Ae j d. Ae e. eT A f. ...
... 1. For a general matrix Am×n describe what the following products will provide. Also give the size of the result (i.e. "n × 1 vector" or "scalar"). a. Ae j b. eiT A c. eiT Ae j d. Ae e. eT A f. ...
PUSD Math News – Mathematics 1 Module 8: Connecting Algebra
... Image- http://study.com/academy/lesson/what-is-slope-intercept-formdefinition-equation-examples.html ...
... Image- http://study.com/academy/lesson/what-is-slope-intercept-formdefinition-equation-examples.html ...
Introduction to systems of linear equations
... If a linear system is consistent, then the solution contains either • a unique solution (when there are no free variables) or • infinitely many solutions (when there is at least one free variable). Example 27. For what values of h will the following system be consistent? 3x1 −9x2 = 4 −2x1 +6x2 = h S ...
... If a linear system is consistent, then the solution contains either • a unique solution (when there are no free variables) or • infinitely many solutions (when there is at least one free variable). Example 27. For what values of h will the following system be consistent? 3x1 −9x2 = 4 −2x1 +6x2 = h S ...
THE PROBABILITY OF CHOOSING PRIMITIVE
... Definition 4. A matrix B ∈ Zp×q is in Hermite normal form if (1) Bij = 0 for all j > i, (2) Bii > 0 for all i, and (3) 0 ≤ Bij < Bii for all j < i. Given any integer matrix B of full row rank, there exists a unimodular matrix U such that BU is in Hermite normal form (see, e.g., [5]; U will not, in g ...
... Definition 4. A matrix B ∈ Zp×q is in Hermite normal form if (1) Bij = 0 for all j > i, (2) Bii > 0 for all i, and (3) 0 ≤ Bij < Bii for all j < i. Given any integer matrix B of full row rank, there exists a unimodular matrix U such that BU is in Hermite normal form (see, e.g., [5]; U will not, in g ...
