A-Level Further Maths: C4-01 Invariance: Invariant Points and Invariant Lines

A-Level Further Maths: C4-01 Invariance: Invariant Points and Invariant Lines - 雙語字幕

In this section we're going to be looking at invariance,

and what I mean by that are either invariant points or a line of invariant points or an invariant line.

So there are a few different situations that we need to consider.

So remember what you're thinking about and what you're imagining is that graph transformations are being represented as matrices.

So these matrices are being applied to these coordinates,

and what are happening is that they could be rotations, they be reflections, they be enlargements or stretches or shears.

So we've looked at a number of different situations, or they could be a combination of those.

Now if a point remains fixed,

so once you've applied the transformation and it has remained in place exactly where it began, we refer to it as an invariant point.

So we have a possibility of an invariant point.

Now there is one point in particular that will remain invariant for all transformations that can be represented as matrices.

And that is the origin.

Because if you have a matrix, a, b, c, d, and you apply that to the coordinates zero, zero, you will get zero, zero.

So it doesn't matter what a, b, c, or d are.

you will get that the origin maps to itself.

So the origin is an invariant point.

Now I also mentioned that you could have a line of invariant points.

So, what that means is that you have a line, well points that are on a line, that once the transformation is applied, remain fixed.

So you could imagine that these all points along the x-axis,

for example, after a reflection in the x-axis, these points will remain precisely where they are.

Okay?

And so this would be a line of invariant.

Okay, and of course there are the points in between as well, not discounting those, okay?

So all of these points would make up this line of invariant points, okay?

So quite often you can think of those as reflection lines, so the mirror lines themselves, all the will remain fixed.

Then last situation is an invariant line.

Now, this is different to a line of invariant points, because an invariant line just says

that points that started on the line remain on the line, but that doesn't necessarily mean that they remain fixed, okay?

So we're not just talking about a line of invariant points here, okay?

So for example, this could be your x-axis, and you have points that are on the x-axis which are mapped along it.

So, that could be a stretch parallel to the x axis or in the x direction by factor 2, say for example.

So, the points that are on the line remain on the line, but they're not a line of invariant

points because the points aren't standing still,

they're not staying still, they are moving along the And so, there is this distinction between those two situations.

So, you can kind of think of this situation as a mirror line.

Whereas this, you might think as a stretch in a certain direction.

Now that's not just meaning that those two transformations the only ones that will create a of a vote points and a variant line,

but it gives you something to kind of connect it to.

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