Matrix spaces

From a matrix can be derived several vector spaces, referred to collectively as matrix spaces.

Suppose A is an m×n matrix.

• The column space of A is the subspace of R^m comprising all vectors Ax where x is in Rⁿ.
• The nullspace of A is the subspace of Rⁿ comprising all vectors x such that Ax = 0.
• The row space of A is the column space of A^T.
• The nullspace of A^T is just that.

To find bases of these spaces, apply Gaussian elimination to A, getting a matrix B in row-echelon form:

One basis of the column space of A is this: the set that contains column j of the matrix A just in case column j of the matrix B has a pivot.

One basis of the row space of A is the set comprising the (transposes of the) non-zero rows of the reduced matrix B.

The columns of B without pivots correspond to free variables of the homogeneous linear system

Ax = 0 .

One basis of the nullspace of A is the set containing one vector for each such free variable, namely the solution-vector in which that free variable is 1 and the other free variables are 0.

For a basis of the nullspace of A^T, one just has to apply Gaussian elimination to that matrix. One can then find bases for the column and row spaces of that matrix, which are, respectively, bases for the row and column spaces of A (albeit different bases in general from the ones just described).

The rank of A is the dimension of its column space. The nullity of A is the dimension of its nullspace. Therefore:

• rank(A) + nullity(A) = n (where n is the number of columns of A).
• rank(A) = rank(A^T).
• rank(A) + nullity(A^T) = m (where m is the number of rows of A).

A vector in Rⁿ is in the nullspace of A if and only if it is orthogonal to every vector in the row space of A.

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