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Q8.1-15E

Expert-verified
Found in: Page 437

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# 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 $${\mathbb{R}^n}$$.

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

See the step by step solution

## Step 1: 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, $$S = \left\{ {x:A{\bf{x}} = {\bf{b}}} \right\}$$.

## Step 2:  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 $${\mathbb{R}^n}$$.

## Step 3:  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 $${\mathbb{R}^n}$$.