[3Blue1Brown] Linear transformations and matrices

-This post was written for arranging content of Essence of Linear Algebra series

   that uploaded by 3Blue1Brown channel in Youtube. -

 

The word, 'transformation' is meaning 'function'.

Takes input and spits out an output each one

 

In Lenear Algebra, taking some vector and spiting out some vector

and this case has something different point.

 

Generally, the relation between inputs and outputs of a function is not really close.

But relation between input vector and output vector is some kind of transformation.

It is expressed by the movement of the vector like this.

But transformation in Linear Algebra is Linear.

It means Lines remain lines, Origin must be remaining fixed, and Grid lines remain parallel and evenly spaced.

 

I said "The word, 'transfrmation' is meaning 'function'."

then what function do you need to transform?

 

Transformation is basically transformation of basis vectors, i-hat and j-hat.

For the combination of two basis vectors is exactly same, so we just need to know how much the basis vector has changed since the before and after transformations.

$transformed \; \vec{v} = -1(transformed \; \mathit{i})+2(transformed \; \mathit{j})$

 

$\vec{v_{t}} =x\begin{bmatrix}1\\-2\end{bmatrix}+y\begin{bmatrix}3\\0\end{bmatrix}$

 

It's common to package of these coordinates called a two-by-two matrix

Using this formular

$\vec{v_{t}}=x\begin{bmatrix}Where\;i\;lands\end{bmatrix}+y\begin{bmatrix}Where\;j\;lands\end{bmatrix}$

 

this corresponds with the idea of adding the scaled versions of our new basis vectors.

If the transformed basis vectors are dependent on each other, it means that the span of the two vectors is squish in one dimension.