Determinants and Standard Orientation

  • Thread starter jakey
  • Start date
  • #1
51
0

Main Question or Discussion Point

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 $R^n$?
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,775
8
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.
 
  • #3
51
0
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!
 

Attachments

  • #4
quasar987
Science Advisor
Homework Helper
Gold Member
4,775
8
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.
 

Related Threads on Determinants and Standard Orientation

Replies
4
Views
5K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
12
Views
3K
  • Last Post
Replies
1
Views
2K
Replies
2
Views
1K
Replies
5
Views
1K
Replies
2
Views
5K
Replies
7
Views
5K
Replies
1
Views
546
Top