Rank

Japanese school year: University year 1

What you learn

Learn the rank of a matrix, defined as the maximum number of linearly independent row or column vectors. It is widely used to determine the solvability and degrees of freedom of linear systems. Prior knowledge of elementary row operations and linear independence is helpful.

Go to practice

Key points

This formula explains the geometric meaning of the rank of matrix $A$ . It equals the dimension of the output space (image $\operatorname{Im}(A)$ ), which matches the number of linearly independent column vectors.

\operatorname{rank}(A) = \dim(\operatorname{Im}(A))

This property shows that taking the transpose $A^T$ does not change the rank of matrix $A$ . In any matrix, the number of linearly independent rows always equals the number of linearly independent columns.

\operatorname{rank}(A^T) = \operatorname{rank}(A)

This inequality gives the maximum possible rank for an $m \times n$ matrix $A$ . The rank can never exceed the smaller value between the number of rows $m$ and columns $n$ .

\operatorname{rank}(A) \le \min(m, n)

This property shows how rank behaves under matrix multiplication. Multiplying matrices never increases the dimension of a space, so the rank of $AB$ cannot exceed the rank of $A$ or $B$ .

\operatorname{rank}(AB) \le \min(\operatorname{rank}(A), \operatorname{rank}(B))

Choose a set to practice.

Practice 1Coming soon