Linear Transformations and Matrices

Related: Matrix

Linear Transformation: Replace the word transformation with “function”. They take in inputs and spits out an output for each one. Transformations take in vectors and spit out another vector.

It’s called a transformation and not a vector because transformations imply movement. If a transformation takes an input vector and spits out an output vector, we imagine that input vector moving over to the output vector.

To understand the transformation as a whole, imagine each possible input vector move over to its corresponding output vector.

Visually speaking a transformation is linear if it has two properties.

• All lines must remain lines without getting curved
• The origin must remain fixed in place

If we’re doing this numerically (say, in programming) what do we do?

It turns out that we only need to record where the two basis vectors, $\hat{i}$ and $\hat{j}$, land— everything else will follow from that.

Its common to package these numbers into a 2x2 grid of numbers called the 2x2 matrix, where you can interpret the columns as the two special vectors where $\hat{i}$ and $\hat{j}$ land.

यसे:

$\left[\begin{array}{cc}{\hat{i}}_x& {\hat{j}}_x\\ {\hat{i}}_y& {\hat{j}}_y\end{array}\right]$

Side tangent from Vish, one way I like to think about this ($\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$) is that a and c go together and b and d go together much like in four-square. 💖

Anyways. MATRIX VECTOR MULTIPLICATION!!!

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=x\left[\begin{array}{c}a\\ c\end{array}\right]+y\left[\begin{array}{c}b\\ d\end{array}\right]=\left[\begin{array}{cc}ax& by\\ cx& dy\end{array}\right]$