Systems of Linear Equations

$Ax=b$ intuition: looking for the a vector(s) that after applying the transform $L_A$ is $b$

A solution is an n-tuple such that. The set of all solutions is a solution set. A system is consistent if the solution set is non-empty.

Homogeneous Systems

• A system is homogeneous if $b=0$
• Let $Ax=b$ be homogeneous system of m eqs and n unknowns
  • Then the solution set $k=N(L_a)$
  • $⟹k$ is a subspace of $F^n$ of dimension $n-\text{rank}(A)$
  • If $m<n$, then there is a nonzero solution
• The solution set of a non homogeneous system includes the solution set of the homogenous solution
  • $k$ solution set for $Ax=b$, $k_H$ solution set of $Ax=0$
  • $k=\{s\}+k_H$

Solving

• Gaussian Elimination
• Cramer’s Rule
• Using Inverses
  • When $A$ is square (system of n equations with n unknowns)
  • $A$ invertible $⟺$ only solution is $A^{-1}b$