- #1
bwpbruce
- 60
- 1
$\textbf{Problem}$
Given $\textbf{v} \ne \textbf{0}$ and $\textbf{p}$ in $\mathbb{R}^n$, the line through $\textbf{p}$ in the direction of $\textbf{v}$ is given by $\textbf{x} = \textbf{p} + t\textbf{v}$. Show that linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps this line onto another line or onto a single point.
$\textbf{My Solution}$:
$\textbf{x} = \textbf{p} + t\textbf{v}$
By Linear Transformation Property:
$T(\textbf{x}) =T(\textbf{p} + t\textbf{v})$
$T(\textbf{x}) =T\textbf{p} + t(T\textbf{v})$
Let $T(\textbf{x}) = \textbf{y}, T\textbf{p} = \textbf{q}, T\textbf{v} = \textbf{w}$
Then $\textbf{y} = \textbf{q} + t\textbf{w}$ is another parametric equation and $\textbf{y}$ is the other line that $\textbf{x}$ maps to except in the case where $\textbf{w} = \textbf{0}$. Then $\textbf{y} = \textbf{q}$
Result:
$\textbf{y} = \textbf{q} + t\textbf{w}$
or
$\textbf{y} = \textbf{q}$
Conclusion:
$\textbf{x} \mapsto \textbf{y}$ by $T$.
Check my solution please?
Given $\textbf{v} \ne \textbf{0}$ and $\textbf{p}$ in $\mathbb{R}^n$, the line through $\textbf{p}$ in the direction of $\textbf{v}$ is given by $\textbf{x} = \textbf{p} + t\textbf{v}$. Show that linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps this line onto another line or onto a single point.
$\textbf{My Solution}$:
$\textbf{x} = \textbf{p} + t\textbf{v}$
By Linear Transformation Property:
$T(\textbf{x}) =T(\textbf{p} + t\textbf{v})$
$T(\textbf{x}) =T\textbf{p} + t(T\textbf{v})$
Let $T(\textbf{x}) = \textbf{y}, T\textbf{p} = \textbf{q}, T\textbf{v} = \textbf{w}$
Then $\textbf{y} = \textbf{q} + t\textbf{w}$ is another parametric equation and $\textbf{y}$ is the other line that $\textbf{x}$ maps to except in the case where $\textbf{w} = \textbf{0}$. Then $\textbf{y} = \textbf{q}$
Result:
$\textbf{y} = \textbf{q} + t\textbf{w}$
or
$\textbf{y} = \textbf{q}$
Conclusion:
$\textbf{x} \mapsto \textbf{y}$ by $T$.
Check my solution please?
Last edited: