Determinants and Standard Orientation

[jakey]

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]

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.

 

[jakey]

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.

If you refer to the attached file, this matix has a positive orientation (and the sign of the determinant is positive) since the direction from (a,b) to (c,d) is counterclockwise, which is the same orientation as R2 (counterclockwise). Thanks!

 

[quasar987]

I was asking that you write out the definition of what it means for a basis of R^2 to have the same (resp. the opposite) orientation as another basis of R^2.

Your little drawing with arrows does not constitute a mathematical definition.

 

[blue_raver22]

I can explain this concept using mathematical principles. The determinant of a matrix is a numerical value that provides information about the transformation represented by the matrix. It is a scalar quantity that is determined by the size and orientation of the rows and columns of the matrix.

The standard orientation of $R^n$ refers to the natural ordering of the basis vectors in $R^n$, which is usually defined as the unit vectors in the positive direction of each coordinate axis. In other words, the standard orientation of $R^n$ is the conventional way of arranging the basis vectors in $R^n$.

Now, when we consider a matrix $A$, the sign of its determinant is dependent on the orientation of its rows with respect to the standard orientation of $R^n$. This means that if the rows of $A$ are arranged in the same order as the standard orientation of $R^n$, then the determinant will be positive. On the other hand, if the rows of $A$ are arranged in the opposite order, the determinant will be negative.

To understand this concept better, we can look at the geometric interpretation of determinants. The determinant of a matrix represents the volume of the parallelepiped formed by the column vectors of the matrix. When the rows of the matrix are arranged in the same order as the standard orientation of $R^n$, the volume of the parallelepiped is positive. This is because the vectors are arranged in a way that forms a right-handed coordinate system, which has a positive volume.

However, if the rows of the matrix are arranged in the opposite order, the volume of the parallelepiped will be negative. This is because the vectors are arranged in a way that forms a left-handed coordinate system, which has a negative volume.

In conclusion, the sign of the determinant of a matrix is determined by the orientation of its rows with respect to the standard orientation of $R^n$. This concept is important in understanding the properties of determinants and their role in linear transformations.