System of linear equations
Dependencies:
Let $F$ be a field. Let $A \in \mathbb{M}_{m, n}(F), X \in \mathbb{M}_{n, 1}, B \in \mathbb{M}_{m, 1}$.
The equation $AX = B$, where $A$ and $B$ are known and $X$ is unknown is called a system of $m$ linear equations in $n$ variables.
This is because expanding the $i^{\textrm{th}}$ row of $AX$ and $B$ gives us:
\[ B[i, 1] = (AX)[i, 1] = \sum_{k=1}^n A[i, k]X[k, 1] \]
which is a linear equation in $n$ variables.
The stacked matrix $[A|B]$ is called the augmented matrix of the system of linear equations.
If $B = 0$, the system is called homogenous. A homogenous system always has a trivial solution $X = 0$. (There may be other solutions where $X \neq 0$)
Dependency for:
- Equations with row equivalent matrices have the same solution set
- Rank of a homogenous system of linear equations
Info:
- Depth: 4
- Number of transitive dependencies: 6