Matrix Rank
linear-algebramatrix-theoryechelon-form
Abstraction: Maximum linearly independent rows or columns in a matrix
Key points:
- Rank is defined as the maximum number of linearly independent column vectors (or equivalently, row vectors) in a matrix
- For an r x c matrix, maximum rank is min(r, c); rank is zero only if the matrix has no elements
- To find rank, transform matrix to row echelon form and count non-zero rows
- A matrix is "full rank" when all its vectors are linearly independent; rank equals min(r, c)
- If a row is a linear combination of other rows it does not contribute to rank (e.g., Row 3 = 3Row1 + 2Row2 leaves rank at 2)
Connections: Matrix Rank · Linear Independence · Row Echelon Form