What does "x-axis invariant" mean?

Greetings,

I'm getting confused about the term "x-axis invariant". A tutorial video which I'm gonna link below states that it means "parallel to the x-axis
". However, a lot of people online seem to think it means "parallel to the y-axis". (When dealing with a 2-D scenario)

(At 6:54) https://www.youtube.com/watch?time_continue=415&v=ILa2kp_Mwkg

Could someone please clarify for me what "x-axis invariant" actually means and why?

Thank you in advance for your time and help.

Reply 1

LesserAndrew

3

I assume it means when you transform the unit square the point on the x axis doesn't move.

Original post by DeadManProp

Greetings,

I'm getting confused about the term "x-axis invariant". A tutorial video which I'm gonna link below states that it means "parallel to the x-axis
". However, a lot of people online seem to think it means "parallel to the y-axis". (When dealing with a 2-D scenario)

(At 6:54) https://www.youtube.com/watch?time_continue=415&v=ILa2kp_Mwkg

Could someone please clarify for me what "x-axis invariant" actually means and why?

Thank you in advance for your time and help.

It means the x-axis stays the same after the transformation. It is unaffected.

Why? Well just take any point on the x-axis and apply the transformation to it. Let the point be

(x,y) = (a,0)

, then applying

\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}

onto it and you get:

\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ 0 \end{pmatrix}

which means the point stays where it was... so every point on the x-axis stays where they are. Hence the x-axis is invariant.

Reply 3

DeadManProp

OP

10

Ahh, I get it. So "x-axis invariant" basically means the x-axis is a line of invariant points for this transformation. Thank you!

Reply 4

Pangol

15

Original post by DeadManProp

Ahh, I get it. So "x-axis invariant" basically means the x-axis is a line of invariant points for this transformation. Thank you!

It could mean that, but it could also mean that every point on the x-axis gets sent to another point on the x-axis. For example, under an enlargement of scale factor 2 centred on the origin, the point (x,0) is sent to the point (2x,0). Not the same point, but still on the x-axis, so the x-axis is invariant here as well.

There's a difference between an invariant line and a line of invariant points. The latter is always the former, but the former need not be the latter.

What does "x-axis invariant" mean?

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