Question

Question: 15. Let \(A\) be an \({\rm{m}} \times {\rm{n}}\) matrix and, given \({\rm{b}}\) in \({\mathbb{R}^m}\), show that the set \(S\) of all solutions of \(A{\rm{x}} = {\rm{b}}\) is an affine subset of \({\mathbb{R}^n}\).

Short answer

It is shown that the set \(S\) of all solutions of \(A{\bf{x}} = {\bf{b}}\)is an affine subset of \({\mathbb{R}^n}\).

Step-by-step solution

Describe the given statement

It is given that \(A\) it is a \(m \times n\) matrix, \({\bf{b}} \in {\mathbb{R}^m}\) and \(S\) is the set of all solutions of \(A{\bf{x}} = {\bf{b}}\).

This implies every point in \(S\) satisfies the system \(A{\bf{x}} = {\bf{b}}\), that is, \(S = \left\{ {x:A{\bf{x}} = {\bf{b}}} \right\}\).

 Use Theorem 3

According to theorem 3, a nonempty set \(S\) is affine if and only if it is a flat. So, we will have to show that \(S\) is a flat.

Assume that \(W\)is the set of all homogeneous solutions of the system\(A{\rm{x}} = 0\). So, \(W\)must be a subspace of \({\mathbb{R}^n}\).

 Use Theorem 6 of solution set of linear systems

If \(p\) be a solution of the system \(A{\bf{x}} = {\bf{b}}\), then the solution set of \(A{\bf{x}} = {\bf{b}}\) is the set of all vectors of the form \(W = p + {v_h}\), where \({v_h}\) is any solution of the homogeneous equation \(A{\bf{x}} = 0\).

Now, as \(S = W + p\), where \(p\) is a solution of the system \(A{\bf{x}} = {\bf{b}}\), so, \(S\) must be a translated set of \(W\). Thus, \(S\) is a flat.

Therefore, the set \(S\) of all solutions of \(A{\bf{x}} = {\bf{b}}\) is an affine subset of \({\mathbb{R}^n}\).