91Ó°ÊÓ

The set of all linear combinations of the columns of matrix A is Col A, or it is called column space of A. Pivot columns are the bases for Col A.

The set of all homogeneous equation solutions\(A{\bf{x}} = 0\)is Nul A, or it is called the null space of A.

It is given that matrix A is of order\(3 \times 5\). By comparing with\(m \times n\), it is obtained that\(m = 3\)and\(n = 5\).

Also, it is given that matrix A has 3 pivot columns, which means that the dimensions of Col A are 3 and the columns of A span\({\mathbb{R}^3}\).

Since\(n = 5\), Nul A is a subspace of 5. Thus, Nul A is not equal to\({\mathbb{R}^2}\), but it is correct that it is in 2-dimensional space.

As there are 3 pivot columns and there are 5 columns in matrix A, it means that there must be 2 free variables. So, the dimension of Nul Ais 2.

Therefore, \[Col A = {\mathbb{R}^3}\] is true but \({\rm{Nul}} A = {\mathbb{R}^2}\) is not true.