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Q21E

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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: In Exercises 21 and 22, mark each statement True or False. Justify each answer. 21. a. A linear transformation from $$\mathbb{R}$$ to $${\mathbb{R}^n}$$ is called a linear functional. b. If $$f$$ is a linear functional defined on $${\mathbb{R}^n}$$ , then there exists a real number $$k$$ such that $$f\left( x \right) = kx$$ for all $$x$$ in $${\mathbb{R}^n}$$. c. If a hyper plane strictly separates sets $$A$$ and $$B$$, then $$A \cap B = \emptyset$$d. If $$A$$ and $$B$$ are closed convex sets and $$A \cap B = \emptyset$$, then there exists a hyper plane that strictly separate $$A$$ and $$B$$.

1. The given statement is false.
2. The given statement is false.
3. The given statement is true.
4. The given statement is false.
See the step by step solution

## Step 1: Use the definition of linear functional

A linear functional on $${\mathbb{R}^n}$$is a linear transformation $$f$$ from $${\mathbb{R}^n}$$ into $$\mathbb{R}$$.

So, statement (a) is false.

## Step 2:  Use the concept of matrix

The system $$f\left( x \right) = Ax$$ is always satisfied with a matrix $$A$$ with a size $$1 \times n$$ for all $$x$$ in $${\mathbb{R}^n}$$.

Similarly, the system $$f\left( x \right) = nx$$ is always satisfied by a point $$n$$ in $${\mathbb{R}^n}$$, where $$x$$ is also in $${\mathbb{R}^n}$$.

So, statement (b) is false.

## Step 3:  Use the definition of strictly separate

According to the definition of strictly separate, the common subset of the sets $$A$$ and $$B$$ is always null.

So, statement (c) is true.

## Step 4:  Use the concept of strictly separating two sets by a hyperplane

According to the concept of strictly separating two sets by a hyperplane, if $$A$$ and $$B$$ are disjoint closed convex sets, but they cannot be strictly separated by a hyperplane (line in$${\mathbb{R}^2}$$ ).

So, the statement in (d) is false.