Review/Outline Recall: If all bunches of d − 1 columns of a... are linearly independent, then the minimum
... Example: Test P = x3 + x + 1 for reducibility, in F2 [x]. Since (deg P )/2 = 3/2, we need only attempt division by polynomials D of degree 1 ≤ d ≤ 3/2. Since degree must be an integer, we need only consider linear (monic) polynomials D. To enumerate these, we have only a choice of constant coefficie ...
... Example: Test P = x3 + x + 1 for reducibility, in F2 [x]. Since (deg P )/2 = 3/2, we need only attempt division by polynomials D of degree 1 ≤ d ≤ 3/2. Since degree must be an integer, we need only consider linear (monic) polynomials D. To enumerate these, we have only a choice of constant coefficie ...
A refinement-based approach to computational algebra in Coq⋆
... directly into the type of objects: for instance, the type of matrices embeds their size, which makes operations like multiplication easy to implement. Also, algorithms on these objects are simple enough so that their correctness can easily be derived from the definition. However in practice, most ef ...
... directly into the type of objects: for instance, the type of matrices embeds their size, which makes operations like multiplication easy to implement. Also, algorithms on these objects are simple enough so that their correctness can easily be derived from the definition. However in practice, most ef ...
Matrix Multiplication Matrix multiplication is an operation with
... right of other pieces have the same number of rows, and pieces that appear above or below other pieces have the same number of columns. So, in the above example, R xx , appearing to the right of the p × 1 column vector rxy , must have p rows, and since it appears below the 1× p row vector ry′x , it ...
... right of other pieces have the same number of rows, and pieces that appear above or below other pieces have the same number of columns. So, in the above example, R xx , appearing to the right of the p × 1 column vector rxy , must have p rows, and since it appears below the 1× p row vector ry′x , it ...
![arXiv:math/0609622v2 [math.CO] 9 Jul 2007](http://s1.studyres.com/store/data/014863311_1-2ba4c2a5c0b25ba9b8b8b609b66b2122-300x300.png)
arXiv:math/0609622v2 [math.CO] 9 Jul 2007
... Define bj to be the column bj = (a1,j , a2,j , . . . , ak,j )T . By linearity of determinants, det(B + iC) is the sum of 2k determinants of matrices M with dimensions k × k, where in column j we can choose to place either bj or i(−1)j+1 b2k+1−j . Given any determinant of this form, we can convert it ...
... Define bj to be the column bj = (a1,j , a2,j , . . . , ak,j )T . By linearity of determinants, det(B + iC) is the sum of 2k determinants of matrices M with dimensions k × k, where in column j we can choose to place either bj or i(−1)j+1 b2k+1−j . Given any determinant of this form, we can convert it ...
