## Matrix Equations

A matrix equation is an equation of the form `Ax = b`, where `A` is a matrix and `x` and `b` are vectors. Matrix equations provide a compact way to represent and solve systems of linear equations.

### Key Concepts
- **Matrix multiplication interpretation**: `Ax` is the linear combination of columns of `A` with weights from `x`
- **Existence**: A solution exists when `b` lies in the column space of `A`
- **Uniqueness**: The solution is unique when `A` has full column rank (all columns are pivot columns)
- **Homogeneous case**: `Ax = 0` always has the trivial solution `x = 0`

### Properties
- If `A` is square and invertible, `Ax = b` has the unique solution `x = A⁻¹b`
- If `A` is not invertible, the system may have no solution or infinitely many solutions
- The set of all solutions to `Ax = 0` forms the null space of `A`

Matrix equations unify the treatment of linear systems and connect algebraic computation with geometric concepts like span, linear independence, and subspaces.