Question

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\(B\).

Short answer

1. The given statement is false.
2. The given statement is false.
3. The given statement is true.
4. The given statement is false.

Step-by-step solution

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.

 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.

 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.

 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.