Ch. 4 Determinants Flashcards
How to find the area of a parallelogram formed from the 2 vectors v, w
area = |det(vw)|
How to find the area of a triangle with vertices a, b, and c
area = 1/2 |det(b-a, c-a)|
How to find the area of a parallelepiped spanned by u, v, w
area = |det(u,v,w)|
Definition of a permutation
A permutation of a non-empty set X is a bijection from X to itself. We put
SX = {bijections: X –> X}
2 cycles are called disjoint if…
their members do not intersect
How to compose 2 cycles σ o τ
Apply τ first and then σ
What is the sign of a permutation
sgn(σ) = (-1)^(k-1)
where k is the number of numbers in that cycle
Then multiply the signs of the individual cycles together to get the overall sign
What is the trace of a square matrix?
The sum of the entries in the diagonal
How to find the adjoint matrix
First find the cofactor matrix, for each element, cover up its row and column, and calculate the determinant of the matrix left, then multiply by (-1)^(rs), where r and s are the row and column of the element
The adjoint is the transpose of this matrix
For a matrix A, the element A12 represents…
The second element across in the first row
det(a,b,c) =
a · (b x c)
(v x w) · (v x w) =
|v x w|^2
Write vector v in polar coordinates
v = R cosθ
sinθ
What is a transposition?
A permutation of 2 elements
