Summary of Linear Algebra

3.2 Norm, Dot Product, and Distance in Rn

• norm : length of vector ||v||
  • Standard Unit vector : vector of 1 length 1/||v||*v
  • distance between u and v : ||u-v||
• Dot Product : Inner Product! > u • v = ||u|| ||v|| cosß (0 ≤ ß ≤ π)
  • u • v > 0 : acute (less than 90˚)
  • u • v < 0 : obtuse
  • u • v = 0 : 90˚
    
    
  • u • v  = u1v1 + u2v2
  • Euclidean inner product
    • u • v  = uvT = vuT
  • cosß = u • v / ||u|| ||v||

 

3.3 Orthogonality

• orthogonal Projection 영사!
• Decompose : w1 + w2 = w1 + (u-w1) = u
• vector component of u along a
  • proja u = u•a / ||a||^ *a
• vector component of u orthogonal to a
  • u - projau = u - u • a / ||a||^ *a 
• norm of projection : ||proja u|| = |u•a| / ||a||

 

 

Chapter 4 : General Vector Spaces

Section 4.1 : Real Vector Spaces

• Vector Space V
  • u , v in V > u + v in V (inherit)
    • u + v = v + u
    • (u+v) + w = u + (v+w)
    • u+0 = 0 +u =u
    • u + (-u) = 0
  • k : any scalar , u in V > ku in V (inherit)
    • k(u+v) = ku + kv
    • (k+m)u = ku + mu
    • k(mu) = (km)u
    • 1u = u

 

Section 4.2 : Subspaces

• Subset W is itself a vector space under V
• Subspace Test : Axiom 1 and 6
• closed : 닫혀있는~
• Solution Spaces of Homogeneous Sytem

 

Section 4.3 : Spanning Sets

• the standard unit vectors Span Rn
• Testing of Spanning
  • det(A) ≠ 0 (if det(A) =0, do not span)
  • Ax=0 : trivial sol 0
  • Ax=b : consistent

 

Section 4.4 : Linear Independence

• Linear Independence : the vector cannot be explained with the other vector.
  • non trivial sol (det(A) =0) > linear dependence
• Linear Independence of the Standard Unit Vectors in Rn
  • k1i + k2j + k3k =0 >> (k1,k2,k3) = (0,0,0)
  • if i and j are dependent, k1 can be express as k1=-k2 > NOT INDEPENDENT
• a basis to R3?
  1. linear independence? > k1i + k2j + k3k =0 : only trivial solution
  2. span? det(A) ≠ 0

 

Section 4.5 : Coordinates and Basis

• S is a basis for V
  : 1. S span V
  2. S is linearly independent
• Uniqueness of Basis Representation
  • v = c1v1 + c2v2 + ••• + cnvn
  • the group of (c1, c2 ••• cn) is unique!
  • coordinates of v : the group of (c1, c2 ••• cn)

 

Section 4.6 : Dimension

• All bases for vector space have the same number of vectors.
• V : finite-dimensional vector space,
  • V > n vectors : Linearly dependent
  • V < n vectors : not span V
• dimension : number of vectors in a basis of V
• Plus / Minus Theorem
  • S U {v} : still linearly independent
  • span(S) = span(S -{v}), if S -{v} remove v from s > still span!

 

Section 4.8 : Row Space, Column Space, and Null Space

• Row space of A : Rn
• Column space of A : Rm
• Null space of  A : null(A)
  
  
• Ax=b is consistent!
• The elementary row operation can change the column space of a matrix
• Elementary row operations don't change dependency relationships between column vectors.
• the number of basis
  • the number of row space basis = the number of  column space basis

 

Section 4.9 : Rank, Nullity, and Fundamental Matrix Spaces

The row space and the column space of a matrix A have the same dimension.

• common dimension(number of vectors) : rank => rank(A)
• demension of null space : nullity => nullity(A)
• mXn matrix
  • rank(A) ≤ min(m,n) for an mXn matrix A
  • rank(A) + nullity(A) =n = dim(V)
    • [number of leading variables] + [number of free variables] = n
      
      
• Equivalent Statements of a n x n matrix.

 

 

Chapter 5 : Eigenvalues and Eigenvectors

Section 5.1 : Eigenvalues and Eigenvectors

• eigenvalue
• eigenvector
• (λI - A)x =0
• det(λI - A)=0
• features of eigen value
  • ∑λ = trace(A)
  • πλ = |A|

 

Section 5.2 : Diagonalization

• B is similar to A
  • B=P-1AP
• Fact
  1. det(P-1AP) = det(P-1)det(A)det(P) =det(A) = πλ
  2. Invertibility : A is invertible, P-1AP is invertible
  3. rank(A) = rank(P-1AP)
  4. nullity(A) = nullity(P-1AP)
  5. trace(A) = trace(P-1AP) = ∑λ
  6. Eigenvalues : A and P-1AP have same eigenvalues.
  7. Eigenspace dimension : dim(A) = dim(P-1AP)
  
  
• Diagonal matrix D, Square matrix A : A > diagonalization
  => D=P-1AP <=> AP = PD
  • An nXn matrix with n distinct eigenvalues is diagonalizable.
    λ1≠λ2≠λ3≠•••≠λn
    
  • P = {p1, p2, •••, pn}

 

Chapter7 : Diagonalizaion and Quadratic Forms

Section 7.1 : Orthogonal Matrices

• orthogonal : its transpose is th same as its inverse!
  A-1 = At
  • AAt = AtA = I
• A is orthogonal
  • the row vector of A > inner product =0
  • the column vector of A > inner product =0
  • transpose A > orthogonal
  • inverse of A > orthogonal
  • product of orthogonal matrices is orthogonal
  • det(A) =1

 

Section 7.2 : Orthogonal Diagonalization

• orthogonally similar : B = PtAP
• A : n x n matrix => D = PtAP
  • orthogonally diagonalizable
  • orthonormal set of n eigenvectors
  • A : symmetric
• Spectral Decomposition
  • A = ∑λiuiuti