Final Flashcards
det(Ei(r))
r
det(Eij)
-1
det(Eij(r))
1
det(In)
1
det(ABC…Z)
det(A)det(B)…det(Z)
Determinant of a Triangular/Diagonal Matrix
product of its diagonal matrix
Determinant: Type of matrices
Only square matrices
rank(A) < n
det(A) = 0
no inverse
Determinant of Transpose
= determinant of the original matrix
Cofactor Expansion
Expand along a row or column
-1^(i+j)
Property of Determinant
Can Commute
Adjoint
Cofactor (without first term)
Transpose
A(adj(A)) = det(A)In 1/det(A)(adj(A)) = A^-1
Powers of Matrices
A^k = QD^kQ^-1
Eigenvalue on Triangular Matrix
Entries on diagonal are eigenvalues
det(A-bIn) = 0
Rank Eigenvalues
For a matrix to have an eigenvalue, rank < n
Type of Matrix for Eigenvalues
Square
Characteristic Polynomial
det(A-xIn)
factor to find eigenvalues
Eigenvectors
(A - yIn)X = 0
find basic solution for all eigenvalues (y)
Diagonalizing
Q invertible
QD = AQ
Make eigenvectors of A the columns of Q
D = 0s with eigenvalue diagonal
If A has n distinct eigenvalues,…
Q is invertible
Similar Matrices
A and diagonalized D
- same determinant, eigenvalues and characteristic polynomials
det(A^-1)
1/det(A)
Vector PQ
[q1-p1]
[q2-p2]
Length of a vector
(x^2 + y^2)^(1/2)
Unit vector of v
v/|v|
Dot Product
Multiply terms that are in the same position and add them together.
vwcos(angle)
Dot Product Properties
- distributive over addition
- commutes with scalar multiples
- dot product commutes
Projection
a parallel to w
b perpendicular to w
a = ((v•w)/|w|^2)w = projwV
b = v - a
Equation of Line through two points
Point plus multiple of vector created by points
Equation of a Plane
Normal vector multiply by change in coordinate
cos(*)=
(u • v)/|u|•|v|
|u x v| =
|u| • |v| sin(*)
v • v =
|v|^2
Equation of line
Point plus multiple of vector
Cross Product (v x w)
det of
[i j k]
[ v ]
[ w ]
Properties of Cross Product
w x v = -(v x w)
u x (v + w) = u x v + u x w
v x (rw) = r(v x w)
v and w are orthogonal to v x w
Projection Transformation Matrix
1/|w|^2 [a^2 ab]
[ab b^2]
Rotation Transformation Matrix
counter-clockwise and clockwise
counterclockwise [cos* -sin*] [sin* cos*] clockwise [cos* sin*] [-sin* cos*]
Definition Linear Transformation
T(v + w) = T(v) + T(w)
T(rv) = rT(v)
Reflection Transformation
Projection transformation with w being equation of line and v being the perpendicular line
A = 2proj - v
Composition Transformations
A: matrix for T
B: matrix for U
T(U(v)) = ABv
Inverse Transformation Conditions
if v =/ w then T(v) =/ T(w)
every vector must have a transformation
Matrix for T^-1
A^-1
Inverse Transformation Rotation
Other direction
Inverse Transformation Projection
None
Subset (U) Conditions
0) 0 vector is in U
i) if u and v is in U then u + v is in U
ii) if u is in U then ru is in U
Null(A)
set of all vectors in R^m such that Ax=0
basic vectors to homogenous system
Im(A)
set of all y vectors in R^n such that Ax=y
span is the set of the columns of the matrix
Eigenspace
Null(A - yIn)
Span
all linear combination of a set of vectors
any span is in R^n
entries in vector
determines the R^n
Linearly Dependent
linear combination equal to zero with at least one parameter not equal to zero
Linearly Independent
all parameters in the linear combination must be equal to 0
det of the vectors =/ 0
Basis
linearly independent span
number of vectors =< n
number of vectors in a basis is the same as all bases
Dimension
of elements in the basis
Rank and Null(A)
m - r = # of vectors in basis of Null(A)
Rank and Col(A)/Row(A)
rank(A) = dimension of Col(A)/Row(A)
Row Space
Span of the rows of A
Basis of Row Space
Rows with leading ones in RREF
Column Space
Span of columns of matrix
Same as Image
Basis of Col(A)
Corresponding columns to the columns with leading 1s in RREF
Determinant of an Inverse
1/det(A)
det(AB) =
det(A)det(B)
det(B^-1)det(A)det(B)
det(A)
- determinants commute since they are simply numbers
A^-1 =
formula
(1/det(A))(adj(A))
Cayley-Hamilton Theorem
put matrix in for x and identity in for constants in the characteristic polynomial to get zero matrix
Conditions for T^-1
T(v) =/ T(w)
every vector must have a transformation
Area of parallelogram
|v||w|sin(*)
Reflection Matrix
(1/a^2+b^2)[b^2-a^2 -2ab]
[ -2ab a^2-b^2]
det(A^h) in a n x n matrix
h^n det(A)
