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jakey
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How do we show that, given a matrix $A$, the sign of the determinant is positive or negative depending on the orientation of the rows of A, with respect to the standard orientation of $R^n$?
quasar987 said:Isn't that the very definition of "the rows making up the columns of A have the same (res. the opposite) orientation as the standard basis" ?
If not, write the definition of orientation. you're using.
Determinants are numerical values that are used to describe and analyze a mathematical object, such as a matrix or a linear transformation. They are used to find solutions to equations, measure the size and shape of objects, and determine whether a system of equations has a unique solution.
The calculation of a determinant depends on the type of object it is being applied to. For a square matrix, the determinant is calculated by multiplying the diagonal elements and subtracting the product of the off-diagonal elements. For a linear transformation, the determinant is calculated using the change in volume of the object after the transformation is applied.
The standard orientation for determinants is the order in which the elements are listed in the matrix. For example, in a 2x2 matrix, the standard orientation is left to right, top to bottom, so the determinant would be calculated as (a*d) - (b*c). This standard orientation allows for consistency and accuracy in calculations.
The standard orientation is important because it ensures that the calculation of determinants is consistent and accurate. It also allows for easier comparison and analysis of matrices and linear transformations. Without a standard orientation, there would be ambiguity and confusion in calculations and results.
Determinants and standard orientation have many real-world applications, such as in physics, engineering, economics, and computer science. They are used to solve systems of equations, determine the stability of structures, analyze financial data, and even in image processing and computer graphics. They are also essential for understanding and solving problems in linear algebra and multivariable calculus.