Video by solving a problem of Homomorphism

http://youtu.be/sE40dp0lrSY

This is the link to go to  homomorphism's exercise.

And with this video and this post
I end the blog of algebra, I hope to approve, even such an abstract subject that sometimes you do not know where to start to solve the exercises on a test.
Always have to try again and again to reach the top.!!

Mathematician’s Interview

Alejandro Sánchez, mathematical engineer. He studied in Madrid and continues its path specializing in statistics. And Algebra teacher devotion and eager to show their skills.

When did you discover you wanted to be a teacher?

 At 14, I explained algebra to my classmates and at 17 started giving my first lessons. Since then I grew fond of the teaching of mathematics and every day more of them own studio.

Are you studying to teach better every day?

. I think the study should not be teaching, I think that teaching is applied, then I study math every day for my own knowledge. Although it is true that it helps me to be more loose in every mathematical resolution. But my studies are usually higher than what I teach.  And I hope to improve my knowledge of mathematics.

 Why you think that students do not approve more math?

Do not hesitate to say, that mathematics learning are not entirely clean and do not analyze in detail all points, mathematics is something deeper than numbers and letters. And many times we try to explain something more difficult than it is.

What strategy and best practices could be implemented to improve math education?As I said not ignore anything an explanation and always explain everything in the easiest way, in order to reach a proper or formal way. But the problem of failure in mathematics is the wrong base.

Very well thank you very much for your answers.

An endomorphism

Is a morphism (or homomorphism) from a mathematical object to itself.

For example, an endomorphism of a vector space V is a linear mapƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.

In any category, the composition of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid, denoted End(X).

An invertible endomorphism of X is called an automorphism. The set of all automorphisms is a subset of End(X) with a group structure, called the automorphism group of X and denoted Aut(X).

In the following diagram, the arrows denote implication:

Any two endomorphisms of an abelian group A can be added together by the rule (ƒ + g)(a) = ƒ(a) + g(a). Under this addition, the endomorphisms of an abelian group form a ring(the endomorphism ring).

 For example, the set of endomorphisms of Zⁿ is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space or module also form a ring, as do the endomorphisms of any object in a preadditive category.

The endomorphisms of a nonabelian group generate an algebraic structure known as a near-ring. Every ring with one is the endomorphism ring of its regular module, and so is a subring of an endomorphism ring of an abelian group, however there are rings which are not the endomorphism ring of any abelian group. 

Gramian matrix

Gramian matrix, Gram matrix or Gramian of a set of vectors V₁,..,V_n; in an inner product space is the Hermitian matrix(a square matrix with complex entries that is equal to its own conjugate transpose) of inner products, whose entries are given by  G_ij= [V_j, V_i]. For finite-dimensional real vectors with the usual Euclidean dot product, the Gram matrix is simply  G=V^TV; where V is a matrix whose columns are the vectors V_k

An important application is to compute linear independence: a set of vectors is linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero.

Characteristic polynomial

Every square matrix is associated with a characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.

Given a square matrix A, we want to find a polynomial whose zeros are the eigenvalues of A. For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a₁, a₂, a₃, etc. then the characteristic polynomial will be: (t-a₁)(t-a₂)(t-a₃)…

This works because the diagonal entries are also the eigenvalues of this matrix.

For a general matrix A, one can proceed as follows. A scalar λ is an eigenvalue of A if and only if there is an eigenvector v ≠ 0 such that:

or

(where I is the identity matrix). Since v is non-zero, this means that the matrix λ I − A is singular (non-invertible), which in turn means that its determinant is 0. Thus the roots of the function det (λ I − A) are the eigenvalues of A, and it is clear that this determinant is a polynomial in λ.

Other important pages.!

This site has lessons on basic algebra topics and techniques, study tips, calculator advice, worksheets, and more.

http://www.algebrahelp.com/

This video compendium offers videos on many topics, such as chemistry, calculus, and ACT test-prep. In particular, you will find a large collection of algebra lessons.

http://www.brightstorm.com/math/algebra/

A Computer system finds and solves algebraic equations in text!!

MIT researchers and colleagues at the University of Washington have developed a computer system that can automatically solve the type of word problems common in introductory algebra classes.

According to Nate Kushman, a graduate student and lead author on the new paper, the work is in the field of “semantic parsing,” or translating natural language into a formal language. “In these algebra problems, you have to build these things up from many different sentences,” he says. “The fact that you’re looking across multiple sentences to generate this semantic representation is really something new.”

The researchers’ system exploits two existing computational tools. One is the computer algebra system Macsyma, developed at MIT in the 1960s, which can distill algebraic equations into a few common templates. The other is a sentence parser, which represents the relationships between words in a sentence as a treelike diagram.

To train their system to map elements in the parsing diagram onto Macsyma’s equation templates, the researchers used hundreds of examples from an online discussion site. The system analyzed hundreds of thousands of “features” of those examples, such as the syntactical relationships between words or words’ locations in different sentences. Kushman also included a few “sanity checks,” such as whether the solution yielded by a particular equation template was a positive integer.

The work could lead to educational tools that identify errors in students’ reasoning and to systems that can solve more complicated problems in geometry, physics, and finance.