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Q21E

Expert-verifiedFound in: Page 437

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 **\(B\)**.**

- The given statement is false.
- The given statement is false.
- The given statement is true.
- The given statement is false.

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

So, statement (a) is false.

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.

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

So, statement (c) is true.

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.

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