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- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/20%3A_Vector_Spaces/20.02%3A_SubspacesLet \(V\) be a vector space over a field \(F\text{,}\) and \(W\) a subset of \(V\text{.}\) Then \(W\) is a subspace of \(V\) if it is closed under vector addition and scalar multiplication; that is, i...Let \(V\) be a vector space over a field \(F\text{,}\) and \(W\) a subset of \(V\text{.}\) Then \(W\) is a subspace of \(V\) if it is closed under vector addition and scalar multiplication; that is, if \(u, v \in W\) and \(\alpha \in F\text{,}\) it will always be the case that \(u + v\) and \(\alpha v\) are also in \(W\text{.}\)
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/22%3A_Finite_Fields/22.05%3A_Additional_Exercises-_Error_Correction_for_BCH_CodesIf errors have occurred in bits \(a_1, \ldots, a_k\text{,}\) then \(w(t) = c(t) + e(t)\text{,}\) where \(e(t) = t^{a_1} + t^{a_2} + \cdots + t^{a_k}\) is the error polynomial. From \(w(t)\) we can com...If errors have occurred in bits \(a_1, \ldots, a_k\text{,}\) then \(w(t) = c(t) + e(t)\text{,}\) where \(e(t) = t^{a_1} + t^{a_2} + \cdots + t^{a_k}\) is the error polynomial. From \(w(t)\) we can compute \(w( \omega^i ) = s_i\) for \(i = 1, \ldots, 2r\text{,}\) where \(\omega\) is a primitive \(n\)th root of unity over \({\mathbb Z}_2\text{.}\) We say the syndrome of \(w(t)\) is \(s_1, \ldots, s_{2r}\text{.}\)
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/13%3A_The_Structure_of_Groups/13.08%3A_Sage_ExercisesThere are no Sage exercises for this chapter.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/19%3A_Lattices_and_Boolean_Algebras/19.06%3A_Programming_ExercisesA Boolean or switching function on \(n\) variables is a map \(f : \{O, I\}^n \rightarrow \{ 0, I\}\text{.}\) A Boolean polynomial is a special type of Boolean function: it is any type of Boolean expre...A Boolean or switching function on \(n\) variables is a map \(f : \{O, I\}^n \rightarrow \{ 0, I\}\text{.}\) A Boolean polynomial is a special type of Boolean function: it is any type of Boolean expression formed from a finite combinatio4n of variables \(x_1, \ldots, x_n\) together with \(O\) and \(I\text{,}\) using the operations \(\vee\text{,}\) \(\wedge\text{,}\) and \('\text{.}\) The values of the functions are defined in Table \(19.33\). Write a program to evaluate Boolean polynomials.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/10%3A_Normal_Subgroups_and_Factor_Groups/10.06%3A_Sage_ExercisesCompare this with the time needed to run the .is_simple() method and realize that there is a significant amount of theory and cleverness brought to bear in speeding up commands like this. (It is possi...Compare this with the time needed to run the .is_simple() method and realize that there is a significant amount of theory and cleverness brought to bear in speeding up commands like this. (It is possible that your Sage installation lacks GAP's “Table of Marks” library and you will be unable to compute the list of subgroups.)
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/12%3A_Matrix_Groups_and_Symmetry/12.07%3A_Sage_ExercisesThere are no Sage exercises for this chapter.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/05%3A_Permutation_Groups/5.05%3A_SageThe second version of \(\sigma\) is a list of “tuples”, which requires a lot of commas and these must be enclosed in a list. (A tuple of length one must be written like (4,) to distinguish it from usi...The second version of \(\sigma\) is a list of “tuples”, which requires a lot of commas and these must be enclosed in a list. (A tuple of length one must be written like (4,) to distinguish it from using parentheses for grouping, as in 5*(4).) The third version uses the “bottom-row” of the more cumbersome two-row notation introduced at the beginning of the chapter — it is an ordered list of the output values of the permutation when considered as a function.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/16%3A_Rings/16.08%3A_Reading_QuestionsWhat is the fundamental difference between groups and rings? Give two characterizations of an integral domain. Provide two examples of fields, one infinite, one finite. Who was Emmy Noether? Speculate...What is the fundamental difference between groups and rings? Give two characterizations of an integral domain. Provide two examples of fields, one infinite, one finite. Who was Emmy Noether? Speculate on a computer program that might use the Chinese Remainder Theorem to speed up computations with large integers.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/09%3A_IsomorphismsMany groups may appear to be different at first glance, but can be shown to be the same by a simple renaming of the group elements. For example, \({\mathbb Z}_4\) and the subgroup of the circle group ...Many groups may appear to be different at first glance, but can be shown to be the same by a simple renaming of the group elements. For example, \({\mathbb Z}_4\) and the subgroup of the circle group \({\mathbb T}\) generated by \(i\) can be shown to be the same by demonstrating a one-to-one correspondence between the elements of the two groups and between the group operations. In such a case we say that the groups are isomorphic.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/16%3A_RingsUp to this point we have studied sets with a single binary operation satisfying certain axioms, but we are often more interested in working with sets that have two binary operations. For example, one ...Up to this point we have studied sets with a single binary operation satisfying certain axioms, but we are often more interested in working with sets that have two binary operations. For example, one of the most natural algebraic structures to study is the integers with the operations of addition and multiplication. If we consider a set with two such related binary operations satisfying certain axioms, we have an algebraic structure called a ring.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/01%3A_Preliminaries/1.03%3A_Reading_QuestionsWhat do relations and mappings have in common? What makes relations and mappings different? State carefully the three defining properties of an equivalence relation. In other words, do not just name t...What do relations and mappings have in common? What makes relations and mappings different? State carefully the three defining properties of an equivalence relation. In other words, do not just name the properties, give their definitions. What is the big deal about equivalence relations? (Hint: Partitions.) Describe a general technique for proving that two sets are equal.
