MAT022A final exam review session

MAT022A

• $(A\cdot B{)}^{-1}=B^{-1}\cdot A^{-1}$, notice how the order is inverted
• find solution for $A\cdot \overline{x}=\overline{b}$
  • find rref of A
  • find special solutions by finding $\mathrm{rref}(A)\cdot \overline{x}=\overline{0}$
    • Find one special solution for each free variable by setting the other free variables to 0. Repeat for all free variables
  • find a particular solution by setting free variables to 0, which means free variables should just be the corresponding entry in the augmented column
  • the set of all solutions is particular solution + s times special solution 1 + t times solution 2 + …, where s,t,… represents any real number
• find (basis for) nullspace of A
  • find rref of A
  • find special solutions of $A\cdot \overline{x}=\overline{0}$
  • special solutions form a basis over the nullspace of A
• find (basis for) column space of A
  • find rref of A
  • check which columns of rref contains pivots; the corresponding columns in the original A form a basis over the column space
• find (basis for) rowspace of A
  • find rref of A
  • rref rows that contain a pivot forms a basis over the rowspace (OK to use rref rows)
• check if a set of vectors are linearly independent
  • method 1: rref
    • combine vectors into a matrix A
    • find rref of A and see if the rank is equal to n
  • method 2: see if the only linear combination that forms $\overline{0}$ is the trivial all zeros solution
    • write the vector equation out
    • convert vector equation to a system of equations
    • solve system of equations and see if all coefficients are 0 (LI).
• Find least squares solution to $A\cdot \overline{x}=\overline{b}$
  • Solve $A^T\cdot A\cdot \overline{x}=A^T\cdot \overline{b}$
  • Multiply both sides by $(A^T\cdot A{)}^{-1}$
  • $\overline{x}=(A^T\cdot A{)}^{-1}\cdot A^T\cdot \overline{b}$