Further Maths - Linear Transformations
What is a linear transformation?
A transformation that only involves linear terms in $x$ and $y$.
\[\left(\begin{matrix}x \\ y\end{matrix}\right) \mapsto \left(\begin{matrix}x + y \\ y - 1\end{matrix}\right)\]
Is that a linear transformation?
No because it’s a translation.
\[\left(\begin{matrix}x \\ y\end{matrix}\right) \mapsto \left(\begin{matrix}2x - y \\ x + y\end{matrix}\right)\]
Is that a linear transformation?
Yes.
\[\left(\begin{matrix}x \\ y\end{matrix}\right) \mapsto \left(\begin{matrix}2y \\ -x^2\end{matrix}\right)\]
Is that a linear transformation?
No because it involves $x^2$ terms.
If something is a linear transformation of
\[\left(\begin{matrix}x \\ y\end{matrix}\right)\]
, how could you write the resulting matrix?
How could you write the transformation
\[\left(\begin{matrix}x \\ y\end{matrix}\right) \mapsto \left(\begin{matrix}ax + by \\ cx + dy\end{matrix}\right)\]
as matrix multiplication?
What is the name for the transformed version of something?
The image.
What is true for any linear transformation?
It always maps the origin onto itself.
What is true about how any linear transformation can be represented?
It can be represented by a matrix.
How could you write the transformation
\[\left(\begin{matrix}x \\ y\end{matrix}\right) \mapsto \left(\begin{matrix}2y + x \\ 3x\end{matrix}\right)\]
as matrix multiplication?
How can you tranform multiple points using a transformation matrix?
Multiply the coordinate matrix by the transformation matrix.
How could you represent the transformation that does nothing as a matrix?
What is the transformation matrix for an enlargement by factor $5$?
A reflection in the x-axis does this to a coordinate: $(x, y) \mapsto (-x, y)$. How could you write this as a transformation matrix?
The transformation matrix that does nothing is also known as…?
The identity matrix.
The transformation matrix that leaves the unit vectors unchanged is?
What property does matrix multiplication NOT have that normal multiplication does have?
Commutativity.
What property does matrix multiplication ACTUALLY have that normal multiplication also has?
Associativity.
I multiply matricies $A B C$. Because of associativity, what can be said?
As long as they stay next to each other, the order doesn’t matter.
If the order of a set of transformations does not matter, what can be said about the order of the matrix multiplication?
It also doesn’t matter.
How can you combine multiple transformations together in terms of matricies?
Multiply the transformation matricies together.
I have a set of coordinates in a matrix $M$. I do transformations transform it by multiplying by matrix $A$ followed by another tranformation matrix $B$. How could you write the overall transformation?
I multiply a coordinate matrix $M$ by $A$ and $B$ like so: $B A M$. How could I rewrite this?
If you rewrite a transformation $B A M$ as $(B A) M$, what trick does this reveal?
That you can perform multiple transformations by multiplying the matrix by the transformation matricies multiplied together.
I do 4 transformations to $M$ in the order $A \to B \to C \to D$. How could I write this?
If you want to transform a matrix $M$ by a transformation matrix $A$, where does the transformation matrix go?
On the left:
\[A M\]How can you think about transformation matricies in terms of their effects on the unit vectors $\i$ and $\j$?
They define new values for the unit vectors $\i$ and $\j$.
What is the unit vector $\i$?
What is the unit vector $\j$?
By thinking about how matricies define new values for the unit vectors $\i$ and $\j$, what is the transformation matrix that swaps coordinates?
What are points that map onto themselves called?
Invariant points.
Since the origin is always left unchanged after a linear transformation, what can it be called?
An invaritant point.
What are lines that map onto themselves called?
Invariant lines.
Why aren’t all points on an invariant line invariant points?
Because the points themselves can move but just be mapped onto a different point on the line. The line itself does not change.
What happens to the coordinate $(x, y)$ if you reflect in the $x$-axis?
What happens to the coordinate $(x, y)$ if you reflect in the $y$-axis?
Because a reflection in the $x$-axis maps
\[(x,y) \mapsto (x, -y)\]
, how could you write the transformation matrix?
Because a reflection in the $y$-axis maps
\[(x,y) \mapsto (-x, y)\]
, how could you write the transformation matrix?
What are the invariant points for a reflection in the $x$-axis?
All points on the $x$-axis are invariant.
What are the invariant lines for a reflection in the $x$-axis?
- The $x$-axis itself
- Any straight line $y=k$
What are the invariant points for a reflection in the $y$-axis?
All points on the $y$-axis are invariant.
What are the invariant lines for a reflection in the $y$-axis?
- The $y$-axis itself
- Any straight line $x=k$
Because a reflection in the line $y = x$ maps
\[(x,y) \mapsto (y, x)\]
, how could you write the transformation matrix?
What are the invariant points for a reflection in the line $y = x$?
Every point on $y = x$.
What is the obvious invariant line for a reflection in the line $y = x$?
The line $y = x$.
What are the not-so-obvious invariant lines for a reflection in the line $y = x$?
Because a reflection in the line $y = -x$ maps
\[(x,y) \mapsto (-y, -x)\]
, how could you write the transformation matrix?
What are the invariant points for a reflection in the line $y = -x$?
Every point on $y = -x$.
What is the obvious invariant line for a reflection in the line $y = -x$?
The line $y = -x$.
What are the not-so-obvious invariant lines for a reflection in the line $y = -x$?
What two vectors should you draw when creating a unit square to work out a transformation?
Visualise the effect on the unit square the reflection in the line $y = -x$?
Visualise the effect on the unit square the reflection in the line $y = -x$?
What is the matrix for a rotation of $\theta$ anticlockwise?
What is the matrix for a rotation of $\theta$ clockwise?
For a rotation transformation, if $0^{\circ} < \theta < 360^{\circ}$, what is the invariant point?
The origin.
For a rotation transformation where $\theta \neq 180^{\circ}$, what are the invariant lines?
There are no invariant lines.
For a rotation transformation where $\theta = 180^{\circ}$, what are the invariant lines?
Any line passing through the origin.
What is the matrix for a stretch with scale factor $a$ and $b$?
For a stretch
\[\left( \begin{matrix} a \& 0 \\\\ 0 \& b \end{matrix} \right)\]
What is the scale factor for the stretch parallel to the $x$ axis?
$a$
For a stretch
\[\left( \begin{matrix} a \& 0 \\\\ 0 \& b \end{matrix} \right)\]
What is the scale factor for the stretch parallel to the $y$ axis?
$b$
For any stretch
\[\left( \begin{matrix} a \& 0 \\\\ 0 \& b \end{matrix} \right)\]
, what are the invariant lines?
The $x$ and $y$ axies.
What is the matrix for a stretch of factor $a$ parallel to the $x$ axis only?
A stretch parallel to the $x$-axis is
\[\left( \begin{matrix} a \& 0 \\\\ 0 \& 1 \end{matrix} \right)\]
What are the invariant points?
The points on the $y$-axis.
A stretch parallel to the $x$-axis is
\[\left( \begin{matrix} a \& 0 \\\\ 0 \& 1 \end{matrix} \right)\]
What are the invariant lines?
Any line $x = k$.
What is the matrix for a stretch of factor $b$ parallel to the $x$ axis only?
A stretch parallel to the $y$-axis is
\[\left( \begin{matrix} 1 \& 0 \\\\ 0 \& b \end{matrix} \right)\]
What are the invariant points?
The points on the $x$-axis.
A stretch parallel to the $y$-axis is
\[\left( \begin{matrix} 1 \& 0 \\\\ 0 \& b \end{matrix} \right)\]
What are the invariant lines?
Any line $y = k$.
A stretch parallel to the $x$-axis means what in practical terms?
A horizontal stretch.
A stretch parallel to the $y$-axis means what in practical terms?
A vertical stretch.
For a transformation, what does the determinant tell you?
The area scale factor.
What does a negative determinant mean for a transformation matrix?
The shape flipped over during the transformation.
Is
\[\left( \begin{matrix} a \& 0 \\\\ 0 \& 1 \end{matrix} \right)\]
a stretch parallel to the $x$-axis or to the $y$-axis?
The $x$-axis.
Is
\[\left( \begin{matrix} 1 \& 0 \\\\ 0 \& b \end{matrix} \right)\]
a stretch parallel to the $x$-axis or to the $y$-axis?
The $y$-axis.
Is
\[\left( \begin{matrix} b \& 0 \\\\ 0 \& 1 \end{matrix} \right)\]
a stretch parallel to the $x$-axis or to the $y$-axis?
The $x$-axis.
Is
\[\left( \begin{matrix} 1 \& 0 \\\\ 0 \& a \end{matrix} \right)\]
a stretch parallel to the $x$-axis or to the $y$-axis?
The $y$-axis.
2020-12-07
What are the dimensions of a $3$-d transformation matrix?
For a transformation
\[\left( \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right)\]
what is the unit vector
\[\left( \begin{matrix} 1 \\ 0 \\ 0\end{matrix} \right)\]
mapped to?
If a transformation matrix has a zero determinant, what must that mean will happened to a shape being transformed?
A shape will be compressed into one dimension or a point.
What happens to some of the information in a matrix when multiplying by a singular matrix?
It is lost.
You can lose information by multiplying by a singular matrix. What operation means you lose information in normal arithmetic?
Multiplying by $0$.
By thinking about the determinant as an area scale factor, why must a zero determinant mean the matrix has no inverse?
Because all of space is compressed into one dimension, there is no inverse which can “uncompress” space.
A transformation maps all of space onto a line. Why can’t this transformation have an inverse?
You can’t just uncompress the space seeing as there will be multiple inverses for one point on the line.
If a transformation increases area by a factor of $5$, what must be true about the inverse of that transformation?
It decreases the area by a factor of $5$.
The determinant of a matrix is $6$. What is the determinant of the inverse of that matrix?
Why when finding the inverse of a matrix do you multiply by $\frac{1}{\text{det}(M)}?
Because if the transformation scales an area by $\text{det}(M)$ its inverse should do the opposite.
What happens to the coordinate $(x, y, z)$ for a reflection in the plane $x = 0$?
What happens to the coordinate $(x, y, z)$ for a reflection in the plane $y = 0$?
What happens to the coordinate $(x, y, z)$ for a reflection in the plane $z = 0$?
What is the transformation matrix for a reflection in the plane $x = 0$?
What is the transformation matrix for a reflection in the plane $y = 0$?
What is the transformation matrix for a reflection in the plane $z = 0$?
What is the easy way of remembering the matrix for a reflection in the plane $x, y$ or $z$?
It’s the identity matrix but you use $-1$ for the part you’re reflecting.
How would you write the coordinate $(x, y, z)$ as a matrix?
What is the matrix for a rotation of $\theta$ anticlockwise about the $x$-axis in three dimensions?
\[\left(\begin{matrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{matrix}\right)\]
What does this matrix describe?
A rotation of $\theta$ anticlockwise about the $x$-axis.
\[\left(\begin{matrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{matrix}\right)\]
What does this matrix describe?
A rotation of $\theta$ anticlockwise about the $y$-axis.
\[\left(\begin{matrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{matrix}\right)\]
What does this matrix describe?
A rotation of $\theta$ anticlockwise about the $z$-axis.
What is the matrix for a rotation of $\theta$ anticlockwise about the $y$-axis in three dimensions?
What is the matrix for a rotation of $\theta$ anticlockwise about the $z$-axis in three dimensions?
What’s the easy way to remember the $3$-d matricies for a rotation of $\theta$ anticlockwise about the $x, y$ or $z$ axis?
- Pretend you’re doing the minor of the $x, y$ or $z$ part of the coordinate matrix
- Fill in what you’d cross out with the $2$-d rotation matrix
- Exception for the $y$-axis!
What’s the exception for a $3$-d rotation of $\theta$ anticlockwise about the $x, y$ or $z$ axis?
If it’s the $y$-axis you use the $2$-d clockwise rotation matrix instead.
What’s the matrix for a shear parallel to the $x$-axis?
What’s the matrix for a shear parallel to the $y$-axis?
\[\left(\begin{matrix} 1 & \lambda \\ 0 & 1 \end{matrix}\right)\]
What does this matrix describe?
A shear parallel to the $x$-axis.
\[\left(\begin{matrix} 1 & 0 \\ \lambda & 1 \end{matrix}\right)\]
What does this matrix describe?
A shear parallel to the $y$-axis.
\[\left(\begin{matrix} 1 \& \lambda \\\\ 0 \& 1 \end{matrix}\right)\left(\begin{matrix} x \\\\ y \end{matrix}\right)\]
What is the result of this matrix multiplication?
\[\left(\begin{matrix} 1 \& 0 \\\\ \lambda \& 1 \end{matrix}\right)\left(\begin{matrix} x \\\\ y \end{matrix}\right)\]
What is the result of this matrix multiplication?
What is the determinant for any shear transformation?
Visualise the result of a shear parallel to the $x$-axis?
Visualise the result of a shear parallel to the $x$-axis?
What does $\lambda$ represent in a shear transformation?
The “stretch” factor.
How can you imagine a shear’s effect on the unit square?
Like the top across the $y$-axis while keeping the bottom in place.
2020-12-08
\[\left(\begin{matrix} k & 0 \\ 0 & k \end{matrix}\right) \left(\begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix}\right)\]
What is the combined matrix transformation?
2020-12-09
What’s the difference between an invariant line and a line of invariant points?
An invariant line means points on the line are mapped to other points on the line. A line of invariant points means every point on the line is invariant.
What’s the first step for finding the line of invariant points for a matrix
\[\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\]
?
Expand
\[\left(\begin{matrix} a & b \\ c & d \end{matrix}\right) \left(\begin{matrix} x \\ y \end{matrix}\right)\]
?
Multiplying out a transformation matrix by a general point $(x, y)$ yields
\[\left(\begin{matrix} ax + by \\ cx + dy \end{matrix}\right)\]
What must this be equal to if you’re working out an invariant point?
If
\[\left(\begin{matrix} ax & + & by \\ cx & + & dy \end{matrix}\right) = \left(\begin{matrix} x \\ y \end{matrix}\right)\]
What two simulataneous equations can you write?
For a point on a line $y = mx + c$, what is the general coordinate matrix?
What’s the first step for finding an invariant line for a matrix
\[\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\]
?
Multiplying out a transformation matrix by a general point $(x, mx + c)$ yields
\[\left(\begin{matrix} 4x + 3mx + 3c \\ -3x - 2mx - 2c \end{matrix}\right)\]
What must be true about the top line of the matrix and the bottom line of the matrix if it still lies on a line $y = mx + c$?
After a lot of hard work, the image point for a transformation has been put back into the straight line formula $y = mx + c$ and simplified. The result is now
\[(m+1)^2 x + (m+1)c = 0\]
For what value of $m$ is it an invariant line?
What are the three steps for finding an invariant line?
- Transforming the general co-ordinate $(x, mx)$
- Substituting back into $y = mx + c$
- Inspecting for what values the equation is true
2022-03-30
How could you reflect a point in a line?
Find where the normal through the point meets the line and then double the vector between the point and the intersection.
2022-05-04
You’ve just substituted a vector back into
\[y = mx + c\]
to get
\[5x + 3mx + 3c = m(4x-2mc -2c) + c\]
What do you do now to either find invariant lines or show that none exist?
Equate coefficients of $x$
\[5 + 3m = 4m - 2m^2\]and look at the discriminant.
2022-05-11
Is the minus sign in the row or the column for an anticlockwise rotation matrix?
In the row.
What’s a stupid mnemonic to remember that the minus sign is in the row for an anticlockwise rotation matrix?
Rotation and Row start with the same letter.
2022-05-22
If a shear is parallel to the $y$-axis, what does that mean about the $y$-axis?
The $y$-axis is a line of invariant points.
If a shear is parallel to the $x$-axis, what does that mean about the $x$-axis?
The $x$-axis is a line of invariant points.