Show that a Parametric Equation Maps To Another Line By Linear Transformation.

[bwpbruce]

$\textbf{Problem}$
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?

 

[Fallen Angel]

Hi,

It's almost correct, but $y$ doesn't need to be a line, read again the statement and fill the degenrate case.

 

[bwpbruce]

Hi,

It's almost correct, but $y$ doesn't need to be a line, read again the statement and fill the degenrate case.

Is that better?

 

[Fallen Angel]

Yes, now is completely correct :D

 

[blue_raver22]

Your solution is correct. You have correctly shown that the parametric equation $\textbf{x} = \textbf{p} + t\textbf{v}$ maps to another line or a single point through the linear transformation $T$. This is a fundamental property of linear transformations, and it is important for understanding how they affect geometric objects in $\mathbb{R}^n$. Good job!