Chapter 8.4, Problem 21E

a)

To determine

To mark:

The given statement, “A linear transformation from R to ${\text{R}}^n$ is called a linear functional” as true or false.

b)

To determine

To mark:

The given statement, “If f is a linear functional defined on ${\text{R}}^n$, then there exists a real number k such that $f\left(\text{x}\right)=k\text{x}$ for all x in ${\text{R}}^n$” as true or false.

c)

To determine

To mark:

The given statement, “If a hyperplane strictly separates sets A and B, then $A\cap B=\varphi$” as true or false.

d)

To determine

To mark:

The given statement, “If A and B are closed convex sets and $A\cap B=\varphi$, then there exists a hyperplane that strictly separates A and B” as true or false.