91Ó°ÊÓ

Thecolumn spaceof matrix A is the set Col A of all linear combinationsof the columns of A.

Col Aequals \({\mathbb{R}^m}\) only when the columns of A span \({\mathbb{R}^m}\). Otherwise, Col A is only a part of \({\mathbb{R}^m}\).

Theorem 5states that A is an invertible \(n \times n\) matrix; then for each b in \({\mathbb{R}^n}\), equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) has the unique solution \({\mathop{\rm x}\nolimits} = {A^{ - 1}}{\mathop{\rm b}\nolimits} \).

Suppose Q is a \(4 \times 4\) matrix and \({\mathop{\rm Col}\nolimits} \,\,Q = {\mathbb{R}^4}\).

When \({\mathop{\rm Col}\nolimits} \,\,Q = {\mathbb{R}^4}\),the columns of Q span \({\mathbb{R}^4}\). According to the invertible matrix theorem, \(Q\) is invertible and equation \(Qx = {\mathop{\rm b}\nolimits} \) has a solution for each b in \({\mathbb{R}^4}\). In addition, each solution is unique as per theorem 5.

Thus, equation \(Qx = {\mathop{\rm b}\nolimits} \) has a unique solution for each b in \({\mathbb{R}^4}\).