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.!

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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.

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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.

ALGEBRA PRESENTATION!

Hi, I'm Sonia, a student of bioengineering, this is my first post of algebra in this two-month course in algebra, I hope someday academic courses to last more than two months to get into the algebra especially and detained.

Meanwhile I will adapt this short time. And I hope to entertain the subject of algebra, which is so abstract, but it can be entertaining.

Cross product.

  And now I'll include a link to a useful tool: Mathway, a site that helps you to solve math problems. https://www.mathway.com/

Double integrals over general regions

Here is my video,
I've done it about the double integration.
I hope you like it.!
   Best regards.

Interview with a mathematician, Jose Ruíz

I will interview a teacher who knew from childhood has always tried to introduce in mathematics , Jose Ruíz. Of which I have a special affection and remembrance.
I was lucky to grow up surrounded by intellectually restless with much enthusiasm to learn and try new things people , was not lucky enough that my parents instill these values ​​me , but I was looking for me and supported me in interesting people. Always trying to move tickets. So I say it is not an excuse , if you want to go to , even if you have more effort , you're going behind a goal if that motivates you .

Q: Does the math are necessary in our daily life? For what reasons should we learn ?

A: Math is necessary in our daily life , for the same reason we need to learn the language to communicate around us , we need mathematics as a language but also is a universal language. Y is used in the same way that we need to read or write . Continually use mathematics to go shopping , to travel , etc.



Q:Many mathematicians believe that mathematics is fun . How is this possible , if most students hate this subject ?

A:Of course they are fun, but I do not sign9ifica that grab effortlessly all the knowledge needed for learning , in the case of mathematics it is clear that this is study and practice many times until you resolve a problem . The literature on mathematics in the recreation field are immense. Today all of us working in this field are disciples of Martin Gardner, whose books belong in the shelves of any math teacher . It was he , for example, who popularized the game of Life, one of the most exciting activities of the statistical module .
How teacher often fails in many teachers of this science ?
Most focused teaching and rote mechanical way , even not being intended that students see it as well , however it is preaching to his disciples. I think you need to focus more on the compression of all the concepts step by step and understand everything. If I understand the formula will be easier to remember .



Q:What do you think of the use of ICT?

A:Are essential and every day are more important to make science more visible and quickly focus them into practice. Although all have not been lucky enough to form them , at that time there was neither a color TV , which later appeared and revolutionized communication.



Q:Anything else to add?

A:I always like to define this verb is so fabulous motivate management is to inspire people , individually or in groups , so as to produce the best possible results .
And to say that work effortlessly gets nowhere .

Thanks for all your answers.