Linear System of Equations

Definition:

• A set of generic linear equations can be expressed as $Ax=y$ where
  • $x\in {\mathbb{R}}^n$ is the vector to be found
  • $y\in {\mathbb{R}}^m$ is the vector of results, each entry is the rhs result of 1 equation
  • $A\in {\mathbb{R}}^{m,n}$ is matrix with coefficients, each row has entries of 1 equation

Solution

• $S\doteq \{x\in {\mathbb{R}}^n:Ax=y\}$
• There can be 0, 1 or infinite solution
• $Ax$ always lies in $\mathcal{R}(A)$, thus $S\ne \varnothing \iff y\in \mathcal{R}(A)$
• The linear equation $Ax=y$ admits a solution if and only if $\text{rank}([Ay])=\text{rank}(A)$
• When such existence condition is satisfiedm the set of all solutions is the affine set $S=\{x=\bar{x}+Nz\}$ where $\bar{x}$ is any vector such that $A\bar{x}=y$ and $N\in {\mathbb{R}}^{n,n-r}$ is a matrix whose columns span the nullspace odf $A$ (hence $AN=0$)
• The system has unique solution if $\text{rank}([Ay])=\text{rank}(A)$ and $\mathcal{N}(A)=0$