Provide an appropriate response
-Which of the following statements are true?
I: Suppose $f : \mathscr { R } ^ { \mathrm { n } } \rightarrow \mathscr { R } ^ { \mathrm { m } }$ is a linear transformation and $\mathrm { S }$ is a convex subset of $\mathcal { R } ^ { \mathrm { n } }$. It follows that the set of images $f ( S )$ is a convex subset of $R ^ { m }$.
II: If $A \subset B$, then conv $A \subset B$.
A) II
B) I
C) Both I and II
D) Neither I or II

Question

Quizplus · Accepted Answer

I is true because linear transformations preserve convexity: if a set $S$ is convex, then for any two points $x, y \in S$, and for any $\lambda \in [0, 1]$, the point $\lambda x + (1 - \lambda)y$ is also in $S$. Since linear transformations map lines to lines (or points in the case of degenerate transformations), the image of any line segment in $S$ under $f$ will remain a line segment in $f(S)$, thus preserving convexity. II is false because the convex hull of $A$ (conv $A$) is the smallest convex set containing $A$, and while it is contained in the smallest convex set containing $B$ if $A \subset B$, it is not necessarily true that conv $A \subset B$ unless $B$ itself is convex.

Provide an Appropriate Response -Which of the Following Statements Are f:Rn→Rmf : \mathscr { R } ^ { \mathrm { n } } \rightarrow \mathscr { R } ^ { \mathrm { m } }f:Rn→Rm

Provide an Appropriate Response
-Which of the Following Statements Are $f : \mathscr { R } ^ { \mathrm { n } } \rightarrow \mathscr { R } ^ { \mathrm { m } }$