Intro
#- This is the topic where linear algebra starts to click to students. Very important
to understand it and visualize it.
- Very hard to describe all of it in words, will try my best. For better understanding
watch the reference video.
Linear Transformation
#- Transformation is a fancy word for function where you give one input and get an output.
- In linear algebra we like to think about transformations that take in a vector and
get output as another vector.
- Transformation suggests that you think using movement.
- If a transformation takes an input vector and gives an output vector we think of
the transformation and input vector moving over to the output vector.
- To think of transformation as a whole we can imagine every possible input vector in
space move over to its corresponding output vector.
- This is very crowded to think about. So, we can visualize every vector not as an
arrow but a point and it would look like:
- Or think of it using a grid(connect all dots using lines):
- And keeping the original grid in the background:
NOTE: This way of representing is quite good as it helps compare the transformation with
the original. - Since, we are working with linear algebra we can limit the transformations to linear
transformations. i.e., transformations which do not make weird curves like:
- A linear transformation has two properties:
- All lines must remain lines without getting curved.
- Origin must remain fixed.
- Examples:
- Last two figures are not linear transformations because the lines get all curvy.
- Although this one keeps the lines straight but the origin moves so this is also
not a linear transformation:
- This one might look like a linear transformation as it keeps the lines straight
and origin fixed:
But it is not because if we pay attention to the diagonal line it makes it curved:
- A better way to think about it in general is Keeping the grid lines parallel and
evenly spaced.
Describing Linear Transformations numerically
#- To do this we only need to record where the two basis vectors \( \hat{\imath} \)
and \( \hat{\jmath} \) go.
- Example: For a vector:
$$
\vec{v} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}
$$
- You can see we only need to know where our basis vectors land to get this.
- You can also see in the diagram why keeping lines parallel is important.
- This also gives us the power to deduce where any vector lands without having to
draw the whole thing.
Matrices and linear transformation
#- The 2D linear transformation is completely
described by just 4 numbers: The 2 coordinates where \( \hat{\imath} \) lands
and the 2 coordinates where \( \hat{\jmath} \) lands.
- So, we can package these coordinates as 2x2 matrix:
$$
\begin{bmatrix}
3 & 2 \\
-2 & 1
\end{bmatrix}
$$
First column is where \( \hat{\imath} \) lands and second column is where \( \hat{\jmath} \) lands lands and second column is where \( \hat{\jmath} \) lands.
- We want to know where that linear transformation takes the vector:
$$
\begin{bmatrix}
5 \\
7
\end{bmatrix}
$$
We will do:
$$
5 \begin{bmatrix} 3 \\ -2 \end{bmatrix} + 7 \begin{bmatrix} 2 \\ 1 \end{bmatrix}
$$
NOTE: Isn’t it matrix multiplication.
- General case:
$$
\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} =
x \begin{bmatrix} a \\ c \end{bmatrix} + y \begin{bmatrix} b \\ d \end{bmatrix} =
\begin{bmatrix} ax+by \\ cx+dy \end{bmatrix}
$$
Here, a and c are where \( \hat{\imath} \) lands and b and d are where \( \hat{\jmath} \) lands.
And x and y are the scalars for the input vector.
References
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