# Further maths help

Hi im currently in a gap year self studying the whole of A level further maths. Im currently on chapter 7 Linear transformations and finding the derivations of the matrices quite difficult. Are the transformations given in the exam, or are we supposed to remember transformations like reflections in axis, or rotations?
One way to not remember all the transformation matrices is to recall
Unparseable latex formula:

[br]\begin{bmatrix}[br]a & c \\[br]b & d [br]\end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}

is the same thing as saying "The matrix maps (transforms) the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ to $\begin{bmatrix} a \\ b \end{bmatrix}$". Similar for $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

So you have two very easy test vectors to check whether your transformation matrix makes sense. Or in reality, you have the two vectors to reconstruct the transformation matrix.

Say for instance, you don't know what the matrix for rotation by 90 degrees (anticlockwise) is. Well, we know $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ should go to $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$, and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ should go to $\begin{bmatrix} -1 \\ 0 \end{bmatrix}$. There you go, the matrix is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.
(edited 7 months ago)
Original post by tonyiptony
One way to not remember all the transformation matrices is to recall
Unparseable latex formula:

[br]\begin{bmatrix}[br]a & c \\[br]b & d [br]\end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}

is the same thing as saying "The matrix maps (transforms) the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ to $\begin{bmatrix} a \\ b \end{bmatrix}$". Similar for $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.

So you have two very easy test vectors to check whether your transformation matrix makes sense. Or in reality, you have the two vectors to reconstruct the transformation matrix.

Say for instance, you don't know what the matrix for rotation by 90 degrees (anticlockwise) is. Well, we know $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ should go to $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$, and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ should go to $\begin{bmatrix} -1 \\ 0 \end{bmatrix}$. There you go, the matrix is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$.

Wow thank you so much! in the exam will i be expected to derive these? can I use the test vectors (1,0) and (0,1) to find the other transformations?
Original post by moosaa0001
Wow thank you so much! in the exam will i be expected to derive these? can I use the test vectors (1,0) and (0,1) to find the other transformations?

I'm not familiar with the exam format, sorry. But I'd bet you will not be given the matrices in the formula sheet or something.
You can find all the linear transformations (so rotation, reflection, dilation... not translation) using (1,0) and (0,1). Try it! It shouldn't take more time than checking the formula sheet, if they were in fact there.
(For instance, try finding the matrix for the reflection along the x-axis, something your textbook has the answer to)
(edited 7 months ago)
Original post by moosaa0001
Hi im currently in a gap year self studying the whole of A level further maths. Im currently on chapter 7 Linear transformations and finding the derivations of the matrices quite difficult. Are the transformations given in the exam, or are we supposed to remember transformations like reflections in axis, or rotations?

I was in a similar position to you and I'd really recommend Mr Bicen on YouTube because the textbooks are rubbish and miss a lot out. His videos are pretty much lessons on every topic and optional modules.
Original post by moosaa0001
Wow thank you so much! in the exam will i be expected to derive these? can I use the test vectors (1,0) and (0,1) to find the other transformations?

These are base vectors (1,0) and (0,1) and you should know how to construct the required transformation.
Are you using Dr Frost?
Original post by Kong, Donkey
I was in a similar position to you and I'd really recommend Mr Bicen on YouTube because the textbooks are rubbish and miss a lot out. His videos are pretty much lessons on every topic and optional modules.

Yes i agree, Mr Bicen's channel has helped alot with previous chapters, however his video for this specific section is very brief.
Original post by Muttley79
These are base vectors (1,0) and (0,1) and you should know how to construct the required transformation.
Are you using Dr Frost?

No, ive just been working through the chapters in the textbook. As well as watching Bicen maths for parts i dont understand. I have had a look at dr frost, but all i have found is some powerpoint presentations which I havent found particularly useful.
Original post by moosaa0001
No, ive just been working through the chapters in the textbook. As well as watching Bicen maths for parts i dont understand. I have had a look at dr frost, but all i have found is some powerpoint presentations which I havent found particularly useful.