5 - Permutations, Vector Spaces and Sub Spaces

• Permutation matrices execute row exchanges
  • Matlab does this to choose higher pivot, improving accuracy
  • Number of permutation matrices possible for a nxn matrix = n! (Arrangement of rows)
  • PA = LU
  • P is orthonormal i.e. P^-1 = P^T and columns are unit vectors

• A^TA is always symmetric i.e. its transpose is equal to itself

• Vector Spaces => set of vectors which satisfy closure property on linear combination
  • Linear combination = k₁v₁ + k₂v₂
    • where k_x = scalars and v_x vectors from that set
  • Closure = After taking linear combination the resultant vector must belong to same set
  • Hence Zero or null vector must be present in any vector space
• Sub Spaces => subset of vector space which satisfies the above closure property
  • R² has R², R¹ through origin, R⁰ i.e origin as subspaces
• Column Space => Span of columns of a matrix

6 - Solutions of Ax=b, Null Space

• If P and L are two subspaces
  • Their union is always a subspace
  • Their intersection may or may not be subspace
• Does Ax=b have a solution for every b?
  • No, only those b have a solution which lie in C(A)
  • Since Ax is nothing but linear combination of columns of A, which is span
• Null Space => set of vectors satisfying Ax = 0
  • Always a subspace
• Solution of Ax = b can be expressed as x = x_p + Cx_s
  • x_p = particular solution
  • x_s = special solution , Ax_s = 0
  • A (x_p + Cx_s) = b

7 - Computing Null Space, Pivot/Free variables, Reduced Row Echelon form

• pivot = first non zero element in every row after elimination
• pivot columns = columns containing pivot
• pivot variables = variables corresponding to pivot columns
• free columns = columns without pivot
• free variables = variables corresponding to free columns

add image of matrix elimination process

• Reduced row echelon form (rref) = pivot should be 1 and elements above pivot should be 0

add image of rref matrix

• R =

  I	F
  0	0

• RN = 0

add image of N = xpivot, xfree and -F, I