Systems of Linear Equations #
A system of linear equations can be written compactly as:
\[ A\mathbf{x}=\mathbf{b} \]This represents:
- a linear transformation applied to an unknown vector (\mathbf{x})
- producing an output vector (\mathbf{b})
Key components #
Coefficient matrix (A) #
(A) contains the coefficients of the variables.
\[ A \in \mathbb{R}^{m\times n} \]Variable vector (\mathbf{x}) #
(\mathbf{x}) contains the unknowns.
\[ \mathbf{x}= \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} \in \mathbb{R}^{n} \]Constant vector (\mathbf{b}) #
(\mathbf{b}) contains constants on the right-hand side.
\[ \mathbf{b}= \begin{bmatrix} b_1\\ b_2\\ \vdots\\ b_m \end{bmatrix} \in \mathbb{R}^{m} \]When does a solution exist? #
The type of solution depends on (A).
Unique solution #
If (A) is square and invertible, then the solution is unique.
\[ \det(A)\neq 0 \] \[ \mathbf{x}=A^{-1}\mathbf{b} \]