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Thread: Determinants

  1. #1 Determinants 
    Forum Ph.D. Heinsbergrelatz's Avatar
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    Determinants, the special number that is associated with matrices. I just want to know, what exactly does the determinants represent? i mean finiding them is surely not a problem, but what role do they play?? transformations?? In school these days, they just tell almost nothing about determinants apart from getting the values of the determinants in , and matrices. I wasn't happy with this, so i went to take more serious study on just determinants, turns out there is a whole bunch of brand new stuff to learn. For example differentiating the determinants, getting a generalized determinant form in a matrices etc.


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  3. #2  
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    I've once got the same problem with determinants as you're experiencing now and I am still not very fond of them, though they are just a function from an n x n - vector space to the real or complex numbers (depending on the entries in the matrices).

    It has the following properties:

    i)If the rank of the matrix is less than n (dimension of the vector space), then it is zero. (det A = 0; det being "determinant, A being an n x n - matrix).

    ii)det E = 1 ....E being the identity matrix.

    iii)det is linear in every column (as well row).
    --> That means it satisfies the properties of a linear function when regarding every single column/row.

    Now that is a definition which puts the determinant into a tangible algebraic object, and it is a definition which is unique to the determinant (which can be proven).

    If you solve a system of linear equations, then the determinant would be the term in the denominator:

    ax + by = c
    dx + ey = f

    => x = (c- by)/a = (f-ey)/d

    => c/a - f/d = by/a - ey/d = y(bd-ae) / ad

    Now if you solve by y, then you've got the determinant - somehow with the wrong sign, it should be ae-bd - in the denominator.

    The same result you'd get if solving by x, and so on in higher-dimensional systems of linear equations.

    It is very useful for determining the solvability (does this word exist? :-D) of a system or for determining the rank of a matrix (that's why it is calles determinant - because you can determin so much by it ).


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  4. #3  
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    Good question. The determinant of an nxn matrix is the (signed) volume of the parallelepiped generated by the n column vectors of the matrix.

    For example, consider a 2x2 matrix. The column vectors v1 and v2 define a parallelogram whose vertices are (0,0), v1, v2, and v1+v2. The absolute value of the determinant of the matrix (v1, v2) is the area of this parallelogram. The sign of the determinant indicates whether the vectors v1 and v2 define the same orientation as the standard coordinate vectors (1,0), (0,1).

    Determinants provide the key to keeping track of how the formulas for volume elements change when we change our coordinate system.
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  5. #4  
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    Quote Originally Posted by salsaonline
    For example, consider a 2x2 matrix. The column vectors v1 and v2 define a parallelogram whose vertices are (0,0), v1, v2, and v1+v2. The absolute value of the determinant of the matrix (v1, v2) is the area of this parallelogram. The sign of the determinant indicates whether the vectors v1 and v2 define the same orientation as the standard coordinate vectors (1,0), (0,1).
    Homework assignment for Heinsbergrelatz and other who want to understand determinants: Prove this statement for yourself.

    Next, Google "Jacobian matrix" to see the use of determinants in calculus.

    You will have to study linear algebra to understand more, but in that study you will find that the determinant is independent of the choice of basis vectors (linear coordinates) and therefore is a property if linear transformations and not just their matrix representation.

    Finally, when you are ready, find a copy of Calculus on Manifolds bt Michael Spivak and see how determinants play the pivotal role in integration noted by salsaonline.

    Warning: In an elementary linear algebra class you will see methods for solving systems of linear equations and for inverting matrices using determinants, and even matrices consisting of determinants of other matrices (the "classical adjoint"). These are of significant theoretical interest. They are also useful tricks if you routinely solve 2x2 or 3x3 systems. But the calculation of determinants involves a great many arithmetic operations, and is computationally intensive, so that solutions by determinant are practical only in quite low dimensions. The importance of determinants lies in the theoretical insight that they bring -- primarily the relationship to "volume" in arbitrary dimension.
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  6. #5  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by salsaonline
    For example, consider a 2x2 matrix. The column vectors v1 and v2 define a parallelogram whose vertices are (0,0), v1, v2, and v1+v2. The absolute value of the determinant of the matrix (v1, v2) is the area of this parallelogram. The sign of the determinant indicates whether the vectors v1 and v2 define the same orientation as the standard coordinate vectors (1,0), (0,1).
    Homework assignment for Heinsbergrelatz and other who want to understand determinants: Prove this statement for yourself.

    Next, Google "Jacobian matrix" to see the use of determinants in calculus.

    You will have to study linear algebra to understand more, but in that study you will find that the determinant is independent of the choice of basis vectors (linear coordinates) and therefore is a property if linear transformations and not just their matrix representation.

    Finally, when you are ready, find a copy of Calculus on Manifolds bt Michael Spivak and see how determinants play the pivotal role in integration noted by salsaonline.

    Warning: In an elementary linear algebra class you will see methods for solving systems of linear equations and for inverting matrices using determinants, and even matrices consisting of determinants of other matrices (the "classical adjoint"). These are of significant theoretical interest. They are also useful tricks if you routinely solve 2x2 or 3x3 systems. But the calculation of determinants involves a great many arithmetic operations, and is computationally intensive, so that solutions by determinant are practical only in quite low dimensions. The importance of determinants lies in the theoretical insight that they bring -- primarily the relationship to "volume" in arbitrary dimension.
    $15. This would be the right book? http://www.alibris.com/booksearch?mt...#search-anchor
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  7. #6  
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    Quote Originally Posted by Arcane_Mathematician
    That's it.
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  8. #7  
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    Quote Originally Posted by salsaonline
    Good question. The determinant of an nxn matrix is the (signed) volume of the parallelepiped generated by the n column vectors of the matrix.

    For example, consider a 2x2 matrix. The column vectors v1 and v2 define a parallelogram whose vertices are (0,0), v1, v2, and v1+v2. The absolute value of the determinant of the matrix (v1, v2) is the area of this parallelogram. The sign of the determinant indicates whether the vectors v1 and v2 define the same orientation as the standard coordinate vectors (1,0), (0,1).

    Determinants provide the key to keeping track of how the formulas for volume elements change when we change our coordinate system.
    oooo i see, interesting. Well anyways thank you for the help.

    Homework assignment for Heinsbergrelatz and other who want to understand determinants: Prove this statement for yourself.

    Next, Google "Jacobian matrix" to see the use of determinants in calculus.

    You will have to study linear algebra to understand more, but in that study you will find that the determinant is independent of the choice of basis vectors (linear coordinates) and therefore is a property if linear transformations and not just their matrix representation.

    Finally, when you are ready, find a copy of Calculus on Manifolds bt Michael Spivak and see how determinants play the pivotal role in integration noted by salsaonline.

    Warning: In an elementary linear algebra class you will see methods for solving systems of linear equations and for inverting matrices using determinants, and even matrices consisting of determinants of other matrices (the "classical adjoint"). These are of significant theoretical interest. They are also useful tricks if you routinely solve 2x2 or 3x3 systems. But the calculation of determinants involves a great many arithmetic operations, and is computationally intensive, so that solutions by determinant are practical only in quite low dimensions. The importance of determinants lies in the theoretical insight that they bring -- primarily the relationship to "volume" in arbitrary dimension.
    YUp i proved Salsaonline's statement, well i stillt think im alittle bit behind if i want to completely understand the Jacobians, as my vector calculus isnt so rigid. Oo.. michael spivak, i have his book "Calculus", just downloaded "calculus on Manifolds" and yup, it seems tough for me right now definitely. Anyway thanks for your advice
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  9. #8  
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    This thread should be useful when I re-read my linear algebra stuff next week.

    Got fed up reading physics and not remembering what a determinant was.

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  10. #9  
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    Solve this 4x4 matrix using Cramer's rule
    (1-a1-a2 ; a1 ; a2 ; 0)
    (b1 ; 1-b1-a2+c2 ; 0 ; a2-c2)
    (b2 ; 0 ; 1-b2-a1+c2 ; a1-c1)
    (0 ; b2+c2 ; b1+c1 ; 1-b1-c1-b2-c2)


    all 4 rows equal zero and the variables to be found (going across the rows) are w, x, y and z.


    When I tried to solve the first determinant I got 4 brackets with 3, 4, 4 and 5 elements in and the multiplying out got too confusing so I am not sure if it was right or not.


    My teacher said Cramer's rule is the easiest way to solve so I want to do it by this method but I've never been taught it and I can't find an example using algebra.


    if there is any thread regarding Algebraic equations so please give me i wil post there my problem...but if you dont mind then please help me...!....
    Last edited by geeknick; November 18th, 2011 at 04:42 AM.
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  11. #10 confused 
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    I am also not getting the same thing What exactly is a determinant!and the replies by those math Geeks is something I couldn't understandCould someone explain it in an easier way?
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  12. #11  
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    That's the definition given in the math booksWell do we use determinants to find the cross product of the given vectors?
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