This Stuff Flashcards
Scalars in subspaces
Scalars fuck with c*u in R unless stated or {u, v} is wide
Subspace H
H = span{v1, … , vn}, for R^n
if zero vector
Null A
All solutions for Ax=0
Col A
ColA = span{ a1, a2, a3 }
vector in NulA or ColA
NulA if Aw=0
ColA if lincomb of vector
Bases
Linindept span
Basis for NullA
REF
x = x1u + x2v
{u, v}
Bases for ColA
Pivot columns in og
Coordinate Vector
B * [x]B = x
dimSubspace
# of columns in basis
Change of Coordinates C <- B
[ b1 b2 | c1 c2 ]
[x]C
P_(C<-B) * x = [x]C
Using EiganValue
x + y = λx
x + y = λy
Using EiganVectors
Ax = λx
EiganSpace
Nul(A - λI)
Characteristic Equation
det(A - λI) = 0
ad - bc = 0
x = EiganValue
Similarity
A and B have the same Characteristic Equation
Inner Product
u * v = uT v
IIuII IIvII cosθ
Length of a vector
||v|| = sqrt(v^2)
||u - v|| = sqrt((u-v)^2)
Unit Vector
if v / IIvII = 1
if v * v = 1
2 Orthogonal Vectors
if u * v = 0
if IIu+vII^2 = IIuII^2 + IIvII^2
Proj A onto B
ŷ = (B*A / A*A) * A
y = ŷ + z
y = All vectors in IR
ŷ = All vectors in W
z = orth to W
Diagonalization
A -> P D P^-1
Orthogonal and Orthonormal
- Orthogonal = set orth from each other
- Orthonormal = unit vector set + orth from each other
Gram Schmidt
v1 = x1
v2 = x2 - (x2v1 / v1v1) v1
Orthonormal Basis
Run Gram Schmidt again
v1 = x1
v2 = x2 - (x2v1 / v1v1) v1
Least Square Normal Equation
A^T Ax = A^T b
Least Square Error
|| b - Ax̄ ||
