Linear Algebra

Linear Algebra

1. Vector Space

   • invertible
     • det(A) ! = 0
   • tr(A)=0
     • tr(A) = sum of diagonal entries
   • Symmetric / Skew Symm
     • Symm: dim(A) = n(n+1) / 2
     • Skew symm: dim(B) = n(n-1) /2
   • Linear Independence
     • AX = 0 Trivial solution
     • col(A) = n
   • Span
     • r(A) = n
     • invertible
     • dim(A) = n ( in Rn)
   • Subspace
     • Subspace Test
     • Basis
       • span{u1,u2,...uk} = V
       • {u1,u2,...uk} is linear independent
   • Else
     • row(A) = rows in RREF of A
     • col(A) = leading col (RREF) in A
     • dimension
       • dim(V) = # of vector in basis
       • r(A) = dim(col(A)) = dim(row(A))
       • dim(Rn) = n
     • image & kernel
       • im(A) = col(A) AX
       • r(A) + dim(null(A)) = n
       • ker(A) = null(A) AX = 0

2. Orthogonal & Orthonormal

   • Orthogonal
   • Orthonormal
   • Fourier Expansion
   • Projection
   • Least Squares

3. Linear Transformation

4. Diagonalization

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