Provide an appropriate response
-Which of the following statements are true?
I: A linear transformation from $\mathscr { R }$ to $R ^ { \mathrm { n } }$ is called a linear functional.
II: The convex hull of a closed set is closed.
A) Neither I or II
B) I
C) II
D) Both I and II

Question

Quizplus · Accepted Answer

I: A linear transformation from $\mathscr { R }$ to $R ^ { \mathrm { n } }$ is called a linear functional. This statement is true because a linear transformation from $\mathscr { R }$ to $R ^ { \mathrm { n } }$ is a function that satisfies the following properties:1. Additivity: $T(x + y) = T(x) + T(y)$ for all $x, y \in \mathscr { R }$.2. Homogeneity: $T(cx) = cT(x)$ for all $x \in \mathscr { R }$ and $c \in \mathbb{R}$.3. Linearity: $T(ax + by) = aT(x) + bT(y)$ for all $x, y \in \mathscr { R }$ and $a, b \in \mathbb{R}$.These properties are the same as the properties of a linear functional, so a linear transformation from $\mathscr { R }$ to $R ^ { \mathrm { n } }$ is a linear functional.II: The convex hull of a closed set is closed. This statement is also true because the convex hull of a closed set is the smallest closed set that contains the given set. Since the given set is closed, the convex hull of the given set is also closed.

Provide an Appropriate Response -Which of the Following Statements Are R\mathscr { R }R

Provide an Appropriate Response
-Which of the Following Statements Are $\mathscr { R }$