Row Space, Column Space & Null Space

Consider A_mXn matrix,

The row vectors, r of A,

r₁ = (a₁₁ .... a_1n), .................... , r_m = (a_m1 ......... a_mn)

The column vectors, c of A,

The subspace span { r₁ ..... r_m } of Rⁿ, where r₁, ......., r_m are the rows of the matrix A, is called the row space of A.

The subspace span { c₁, ...... , c_n }  of R^m, where c_{1, .....,} c_n are the columns of the matrix A, is called the column space of A.

The solution space of the system of homogeneous linear equations Ax = 0 is called the nullspace of A.

To find a basis for the row space of A, the row echelon form of A to be derived, and consider the non-zero rows that result from this reduction. These non-zero rows are linearly independent.

To find a basis for the column space of A, the row echelon form of A to be derived, and consider the pivot columns that result from this reduction. These pivot columns are linearly independent, and that any non-pivot column is a linear combination of the pivot columns.

The dimension of the row space of A is equal to the number of non-zero rows in the row echelon form. The dimension of the column space of A is equal to the number of pivot columns in the row echelon form. The nullity of a matrix is the dimension of its nullspace.

For any matrix A with entries in R, the dimension of the row space is the same as the dimension of the column space.

The rank of a matrix A is equal to the common value of the dimension of its row space and the dimension of its column space.

For any matrix A with entries in R, the sum of the dimension of its column space and the dimension of its nullspace is equal to the number of columns of A, known as the Rank-nullity theorem.