From S.L linear algebra:
Let $V$ be a vector space over $\mathbb{R}$, and let $v,w \in V$. The line passing through $v$ and parallel to $w$ is defined to be the set of all elements $v+tw$ with $t \in R$. The line segment between $v$ and $v+w$ is defined to be the set of all elements $v+tw$ with $0 \leq t \leq 1$. Between what points?
Let $L: V \rightarrow U$ be a linear map. Show that the image under $L$ of a line segment in $V$ is a line segment in $U$. Show that the image of a line under $L$ is either a line or a point. Between what points?
Show that the image of a line under L is either a line or a point.
Formally, I defined some arbitrary line segment $X \in V$ as:
$X=\{v+tw \mid v,w \in W \land 0 \leq t \leq 1 \}$
Then, two axioms of linear mapping are:
- $L(X)=L(X)+L(Y)$, for $Y \in V$.
- $cL(X)=L(cX)=\{cv+ctw \mid v,w \in W \land 0 \leq t \leq 1 \}$, for $c \in \mathbb{R}$.
But the the mapping itself isn't explicitly defined, the only clue is linearity that produces two axioms above (which by my assumption should show that the image under $L$ of a line segment in $V$ is a line segment in $U$).
How can I utilize limited information given from the book above in order to describe the image under $L$ of linear segment in a vector space? Is there any additional information that is a key to the solution?
Thank you!