Question

Suppose A is an m × n matrix of rank r. The nullity of A is __________, and the column space of A is a subspace of __________. Find the dimensions of A.

a) m - r; n

b) n - r; m

c) n; m - r

d) r; m × n

Answer 1

The nullity of matrix A is n - r, and the column space of A is a subspace of ℝ^m. The dimensions of A are m × n.

Step-by-step explanation:

The nullity of matrix A is defined as the dimension of the null space (or kernel) of A, which is given by the number of columns n minus the rank r of the matrix. Therefore, the nullity of A is n - r.

The column space of A, also known as the range of A, is a subspace of ℝm, because the columns of A are composed of m-dimensional vectors. Hence, the column space of A is a subspace of ℝm.

The dimensions of matrix A are simply given by the number of rows m and the number of columns n, which is expressed as m × n.