Core Pure - Linear Transformations Flashcards
What is the new point after a transformation called?
The image
What does a linear transformation mean?
In equations?
All variables in the matrix is in the form ax+by
What is a common mistake in identifying a linear transformation?
X+3 is not a linear transformation
What does a linear transformation always do?
It always maps the origin onto itself
What can all linear transformations be represented as?
A matrix
What is the general form to explain matrices in linear transformations?
What makes this statement true?
What is the notation for a transformed point?
Point S’ has been transformed from point S
Or: look at the image
How can all linear transformations be described?
They can be described on the effect that they have on the unit vectors:
What is the most important thing with 2x2 matrix linear transformations?
What does this infer: T(a+b)
Since T(a+b)=Ta+Tb you can add 2 image vectors to find the 4th vertex after a transformation has been done
When won’t change in a linear transformation?
The origin
What is an invariant point?
points that are mapped onto themselves under the given transformation
What is an invariant line?
lines that each point is mapped back onto the line
They can move along the line so be careful
What is always an invariant point?
The origin
What matrix represents a reflection in the y axis?
(2x2)
When doing a reflection in they y axis, where are the invariant points?
All points on the y axis
When doing a reflection in they y axis, which lines are invariant lines?
X=0 and y=k (for any value of k)
What matrix represents a reflection in the x axis?
When doing a reflection in the x axis, which points are invariant points?
All points on the x axis
When doing a reflection in the x axis, which lines are invariant lines?
The lines: y=0 and x=k (for any value of k)
State the matrices used for a reflection in the x and y axis
What 4 lines do you need to know the matrix for a reflection in?
y and x axis
y=x
y=-x
What matrix represents the reflection in the line y=x?
When doing a reflection in the line y=x, what are the invariant points?
Points on the line y=x
When doing a reflection in the line y=x, what are the invariant lines?
The lines y=x and y=-x+k (for any value of k) are invariant lines
What matrix represents the reflection in the line y=-x?
When doing a reflection in the line y=-x, what are the invariant points?
Points on the line y=-x are invariant points
When doing a reflection in the line y=-x, what are the invariant lines?
The lines y=-x and y=x+k (for any value of k) are invariant lines
How do you show that some line (eg y=3x) is invariant under this transformation?
What is the matrix which represents an anticlockwise rotation through angle , ø, around the origin
What are the 4 reflection matrices that you need to know?
State the matrices
What is extremely important to remember when using the rotation matrix?
It is in an anticlockwise direction
When using the rotation matrix, what is the invariant point?
0,0 (the origin)
When using the rotation matrix, when is there no invariant lines?
When ø≠180
When using the rotation matrix, when are there invariant lines and what are they?
When ø=180
Any line passing through the origin is an invariant line when ø=180
When has an enlargement occurred?
When both sides have been enlarged by the same factor
What do you do if you are asked to describe a stretch?
You need to state the scale factor for both sides
What is the stretch matrix?
Explain the core concept of the stretch matrix?
There is a stretch of scale factor a parallel to the x axis
There is a stretch of scale factor b parallel to they y axis
When is the stretch matrix going to result in an enlargement?
When a=b
What are the invariant lines in the stretch matrix?
What is the invariant point?
The x and y axis are invariant lines
The origin is the invariant point
What will the simpler matrix be for a stretch only parallel to the x axis?
What will the simpler matrix be for a stretch only parallel to the y axis?
For a stretch parallel to the x axis only, what are the invariant points and lines?
For a stretch parallel to the x axis only, points on the y axis are invariant points and any line parallel to the x axis is an invariant line
For a stretch parallel to the y axis only, what are the invariant points and lines?
For a stretch parallel to the y axis only, points on the x axis are invariant points and any line parallel to the y axis is an invariant line
What is the geometry difference between Reflections and stretches?
Reflections preserve their shape and area
Rotations preserve their shape and area
Stretches don’t preserve their shape or area
For a linear transformation represented by matM, what does detM represent?
Scale factor for the change in area
Aka, the area scale factor
Using the determinant ow do you know that a reflection has occurred?
DetM will be negative so the area scale factor will be negative
How can you calculate the area scale factor?
Finding detM
The area scale factor will be the linear scale actor squared
What do you know about a non zero singular 2x2 matrix representing a linear transformation?
It will map any point in the plane onto a straight line through the origin
What matrix represents a reflection in the plane x=0?
What matrix represents a reflection in the plane z=0?
What matrix represents a rotation of angle θ around the x axis?
What matrix represents a rotation of angle θ around the y axis?
What matrix represents a rotation of angle θ around the z axis?
Is the rotation around the x,y and z axis in the clockwise pr anticlockwise direction?
Anticlockwise
How do you calculate the volume of a tetrahedron?
In all cases how is θ measured in a rotation matrix?
In all cases θ is the angle measured anticlockwise when facing towards the origin from the positive side of the axis
What does the matrix PQ represent?
The Matrix PQ represents the transformation of Q (with matrix Q) followed by the transformation of P (with the matrix P)
(Think about reversing the compound transformation)
What is the only time that you will need to recognise that a matrix will be squared?
If C is represented by matM, then C followed by C will be represented by M^2
What are the 3 unit vectors when you are in 3D?
In 3D what does x=0 represent?
They planer of x (a 2D shape not a line)
Have a general idea of what the different planes look like?
What 2 things do you need to remember about matrices?
AB≠BA
How can we use the inverse of a matrix to our advantage?
The bottom bit is the important bit
What do you need to remember to do when describing a rotation?
Say AROUND THE ORIGIN
.
What is the matrix for a reflection in the y axis?
100
0-10
001
What is a tip for reflections in 3D matrices?
The only change is in that reflections unit vector
What is a top for 3D rotation matrices?
That rotations unit vector stays the same
