Linear Systems

$A$ is $m\times n$ matrix

Equivalence

• (d1.7.1) two linear systems are equivalent if and only if they have the same solution set.

Consistency

The following statements are equivalent: (For each $\mathbf{b}\in {\mathbb{R}}^m$)

• $A\mathbf{x}=\mathbf{b}$ is consistent

• (8.6.2) Rouché–Capelli theorem $\rho ([A|\mathbf{b}])=\rho (A)$

• $\text{REF}[A|\mathbf{b}]$ has no row \left[\begin{array}& 0 & \cdots & 0 & b \end{array}\right] in which $b\ne 0$

• $\mathbf{b}$ is a linear combination of the colmuns of $A$

• $\mathbf{b}\in \text{column-space}(A)$

• todo The colmuns of $A$ span ${\mathbb{R}}^m$, then Ax=b for each b

• todo $A$ has a pivot position in every row, then Ax=b for each b

Properties of consistent system: number of soltions:

• if there are no free variables, there is a unique solution
• if there is at least one free variable, there are infinitely many solutions

Homogeneous system

$A\mathbf{x}=0$

• Homogeneous system is consistent
• if $m<n$, then it has infinitely many solutions
• The following statements are equivalent:
  • $A\mathbf{x}=0$ has a nontrivial solution
  • The columns of a matrix $A$ are linearly dependent
  • $A\mathbf{x}=0$ has at least one free variable
• The following statements are equivalent:
  • $A\mathbf{x}=0$ has only the trivial solution
  • The columns of a matrix $A$ are linearly independent

Overdetermined Case (m>n)

• if $m>n$ then the $A\mathbf{x}=\mathbf{b}$ is inconsistent for at least one vector $\mathbf{b}$ in ${\mathbb{R}}^m$

Underdetermined Case (m<n)

• if $m<n$ then for each vector $\mathbf{b}$ in ${\mathbb{R}}^m$ the system $A\mathbf{x}=\mathbf{b}$ is either inconsistent or has infinitely many solutions.

Square System (m=n)

• The following statements are equivalent:

  • $A$ is Invertible
  • For each $\text{b}$, the system $A\text{x}=\text{b}$ has unique solution. ($\text{x}=A^{-1}\text{b}$)
  • There exists $\text{b}$, such that the system $A\text{x}=\text{b}$ has a unique solution
  • $A\mathbf{x}=0⟹\mathbf{x}=0$

• The following statements are equivalent:

  • $A$ is singular
  • $A\mathbf{x}=0$ has non trivial solution

• (4.6.1) Cramer’s rule: in that case, $\mathbf{x}=(x_1,\dots ,x_n)$, $x_i=\frac{\det (A_i)}{\det (A)}$, $i=1,\dots ,n$, where $A_i$ is the matrix formed by replacing the $i$-th column of $A$ by the column vector $\mathbf{b}$.