# Further Maths - Linear Transformations

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/linear-transformations/ · Updated: 2024-08-24 · Tags: further-maths, latex-block-alt

##### 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??
$$
\left(\begin{matrix}ax + by \\\\ cx + dy\end{matrix}\right)
$$

##### 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??
$$
\left( \begin{matrix} a \& b \\\\ c \& d \end{matrix} \right)\left( \begin{matrix} x \\\\ y \end{matrix} \right)
$$

##### 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??
$$
\left( \begin{matrix} 1 \& 2 \\\\ 3 \& 0 \end{matrix} \right)\left( \begin{matrix} x \\\\ y \end{matrix} \right)
$$

##### 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??
$$
\left( \begin{matrix} 1 \& 0 \\\\ 0 \& 1 \end{matrix} \right)
$$

##### What is the transformation matrix for an enlargement by factor $5$??
$$
\left( \begin{matrix} 5 \& 0 \\\\ 0 \& 5 \end{matrix} \right)
$$

##### 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??
$$
\left( \begin{matrix} -1 \& 0 \\\\ 0 \& 1 \end{matrix} \right)
$$

##### The transformation matrix that does nothing is also known as...??
The identity matrix.

##### The transformation matrix that leaves the unit vectors unchanged is??
$$
\left( \begin{matrix} 1 \& 0 \\\\ 0 \& 1 \end{matrix} \right)
$$

##### 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??
$$
B A M
$$

##### I multiply a coordinate matrix $M$ by $A$ and $B$ like so: $B A M$. How could I rewrite this??
$$
(B A) M
$$

##### 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??
$$
D C B A M
$$

##### 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 $\i$??
$$
\left(\begin{matrix} 1 \\\\ 0 \end{matrix}\right)
$$

##### What is the unit vector $\j$??
$$
\left(\begin{matrix} 0 \\\\ 1 \end{matrix}\right)
$$

##### By thinking about how matricies define new values for the unit vectors $\i$ and $\j$, what is the transformation matrix that swaps coordinates??
$$
\left( \begin{matrix} 0 \& 1 \\\\ 1 \& 0 \end{matrix} \right)
$$

##### 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??
$$
(x, -y)
$$

##### What happens to the coordinate $(x, y)$ if you reflect in the $y$-axis??
$$
(-x, y)
$$

##### Because a reflection in the $x$-axis maps $$(x,y) \mapsto (x, -y)$$, how could you write the transformation matrix??
$$
\left( \begin{matrix} 1 \& 0 \\\\ 0 \& -1 \end{matrix} \right)
$$

##### Because a reflection in the $y$-axis maps $$(x,y) \mapsto (-x, y)$$, how could you write the transformation matrix??
$$
\left( \begin{matrix} -1 \& 0 \\\\ 0 \& 1 \end{matrix} \right)
$$

##### 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??
$$
\left( \begin{matrix} 0 \& 1 \\\\ 1 \& 0 \end{matrix} \right)
$$

##### 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 $y = x$??
$$
y = -x + k
$$

##### Because a reflection in the line $y = -x$ maps $$(x,y) \mapsto (-y, -x)$$, how could you write the transformation matrix??
$$
\left( \begin{matrix} 0 \& -1 \\\\ -1 \& 0 \end{matrix} \right)
$$

##### 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 $y = -x$??
$$
y = x + k
$$

##### What two vectors should you draw when creating a unit square to work out a transformation??
$$
\i\\, \&\\, \j
$$

##### ![PHOTO UNIT SQUARE](paste-5499321e5afce6e365b1f6696f72ae9861d5a83c.jpg) Visualise the effect on the unit square the reflection in the line $y = -x$??
![PHOTO UNIT SQUARE REFLECT](paste-47258aba8f743080fb3d4fdef3ce78bb5a17c233.jpg)

##### What is the matrix for a rotation of $\theta$ anticlockwise??
$$
\left( \begin{matrix} \cos\theta \& -\sin\theta \\\\ \sin\theta \& \cos\theta \end{matrix} \right)
$$

##### What is the matrix for a rotation of $\theta$ _clockwise_??
$$
\left( \begin{matrix} \cos\theta \& \sin\theta \\\\ -\sin\theta \& \cos\theta \end{matrix} \right)
$$

##### 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 $b$??
$$
\left( \begin{matrix} a \& 0 \\\\ 0 \& b \end{matrix} \right)
$$

##### 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??
$$
\left( \begin{matrix} a \& 0 \\\\ 0 \& 1 \end{matrix} \right)
$$

##### 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??
$$
\left( \begin{matrix} 1 \& 0 \\\\ 0 \& b \end{matrix} \right)
$$

##### 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??
$$
3 \times 3
$$

##### 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??
$$
\left( \begin{matrix} a \\\\ d \\\\ g \end{matrix}
$$

##### 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??
$$
\frac{1}{6}
$$

##### 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 $x = 0$??
$$
(-x, y, z)
$$

##### What happens to the coordinate $(x, y, z)$ for a reflection in the plane $y = 0$??
$$
(x, -y, z)
$$

##### What happens to the coordinate $(x, y, z)$ for a reflection in the plane $z = 0$??
$$
(x, y, -z)
$$

##### What is the transformation matrix for a reflection in the plane $x = 0$??
$$
\left(\begin{matrix} -1 \& 0 \& 0 \\\\ 0 \& 1 \& 0 \\\\ 0 \& 0 \& 1 \end{matrix}\right)
$$

##### What is the transformation matrix for a reflection in the plane $y = 0$??
$$
\left(\begin{matrix} 1 \& 0 \& 0 \\\\ 0 \& -1 \& 0 \\\\ 0 \& 0 \& 1 \end{matrix}\right)
$$

##### What is the transformation matrix for a reflection in the plane $z = 0$??
$$
\left(\begin{matrix} 1 \& 0 \& 0 \\\\ 0 \& 1 \& 0 \\\\ 0 \& 0 \& -1 \end{matrix}\right)
$$

##### 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??
$$
\left(\begin{matrix} x \\\\ y \\\\ z \end{matrix}\right)
$$

##### What is the matrix for a rotation of $\theta$ anticlockwise about the $x$-axis in three dimensions??
$$
\left(\begin{matrix} 1 \& 0 \& 0 \\\\ 0 \& \cos\theta \& -\sin\theta \\\\ 0 \& \sin\theta \& \cos\theta \end{matrix}\right)
$$

##### $$\left(\begin{matrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{matrix}\right)$$ 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??
$$
\left(\begin{matrix} \cos\theta \& 0 \& \sin\theta \\\\ 0 \& 1 \& 0 \\\\ -\sin\theta \& 0 \& \cos\theta \end{matrix}\right)
$$

##### What is the matrix for a rotation of $\theta$ anticlockwise about the $z$-axis in three dimensions??
$$
\left(\begin{matrix} \cos\theta \& -\sin\theta \& 0 \\\\ \sin\theta \& \cos\theta \& 0 \\\\ 0 \& 0 \& 1 \end{matrix}\right)
$$

##### 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??
$$
\left(\begin{matrix} 1 \& \lambda \\\\ 0 \& 1 \end{matrix}\right)
$$

##### What's the matrix for a shear parallel to the $y$-axis??
$$
\left(\begin{matrix} 1 \& 0 \\\\ \lambda \& 1 \end{matrix}\right)
$$

##### $$\left(\begin{matrix} 1 & \lambda \\ 0 & 1 \end{matrix}\right)$$ 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??
$$
\left(\begin{matrix} x+\lambda y \\\\ y \end{matrix}\right)
$$

##### $$\left(\begin{matrix} 1 \& 0 \\\\ \lambda \& 1 \end{matrix}\right)\left(\begin{matrix} x \\\\ y \end{matrix}\right)$$ What is the result of this matrix multiplication??
$$
\left(\begin{matrix} x \\\\ y + \lambda x \end{matrix}\right)
$$

##### What is the determinant for any shear transformation??
$$
1
$$

##### ![PHOTO UNIT SQUARE PRE SHEAR](paste-1578276847d2d51b5147b93c7b9bd397a997c0ef.jpg) Visualise the result of a shear parallel to the $x$-axis??
![PHOTO UNIT SQUARE SHEARED](paste-fce3c84f532079a7b3490e46957656d8325748a3.jpg)

##### 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??
$$
\left(\begin{matrix} k\cos\theta \& -k\sin\theta \\\\ k\sin\theta \& k\cos\theta \end{matrix}\right)
$$

### 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 $$\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)$$??
$$
\left(\begin{matrix} a \& b \\\\ c \& d \end{matrix}\right) \left(\begin{matrix} x \\\\ y \end{matrix}\right)
$$

##### Expand $$\left(\begin{matrix} a & b \\ c & d \end{matrix}\right) \left(\begin{matrix} x \\ y \end{matrix}\right)$$??
$$
\left(\begin{matrix} ax + \& by \\\\ cx  + \& dy \end{matrix}\right)
$$

##### 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??
$$
 \left(\begin{matrix} x \\\\ y \end{matrix}\right)
$$

##### 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??
$$
ax + by = x
$$

$$
cx + dy = y
$$

##### For a point on a line $y = mx + c$, what is the general coordinate matrix??
$$
 \left(\begin{matrix} x \\\\ mx+c \end{matrix}\right)
$$

##### What's the first step for finding an invariant line for a matrix $$ \left(\begin{matrix} a & b \\ c & d \end{matrix}\right) $$??
$$
\left(\begin{matrix} a \& b \\\\ c \& d \end{matrix}\right) \left(\begin{matrix} x \\\\ mx+c \end{matrix}\right)
$$

##### 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 $y = mx + c$??
$$
\text{bottom} = m(\text{top}) + c
$$

##### 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??
$$
-1
$$

##### What are the three steps for finding an invariant line??
1. Transforming the general co-ordinate $(x, mx)$
2. Substituting back into $y = mx + c$
3. 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.

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