Basis and Rank #
A basis is a minimal set of linearly independent vectors that spans a space.
The dimension of a space is the number of vectors in a basis.
Rank #
The rank of a matrix is the dimension of its column space (number of independent columns).
\[ \text{rank}(A) = \dim(\text{Col}(A)) \]Rank measures the number of linearly independent columns (or rows) of a matrix.
Interpretation #
- High rank → richer information
- Low rank → redundancy
| Condition | Result |
|---|---|
| \( \text{rank}(A) < \text{rank}([A \mid \mathbf{b}]) \) | No solution (Inconsistent system) |
| \( \text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) \) , no free variables | Unique solution |
| \( \text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) \) , free variables exist | Infinitely many solutions |
Key idea:
The rank measures the number of independent equations.
Free variables indicate degrees of freedom in the solution.