$$ \newcommand{\i}{\mathbf{i}} \newcommand{\j}{\mathbf{j}} \newcommand{\cvec}[2]{\begin{pmatrix}#1\\#2\end{pmatrix}} \newcommand{\mat}[4]{\begin{bmatrix}#1 & #2\\#3 & #4\\ \end{bmatrix}} \newcommand{\scvec}[2]{\tiny{\cvec{#1}{#2}}} \newcommand{\smat}[4]{\tiny{\mat{#1}{#2}{#3}{#4}}} \newcommand{\nth}{n^{\text{th}}} $$

Notes from the Essence of Linear Algebra video series by 3blue1brown.

Linear transformations and matrices

A linear transformation is completely specified by

  1. Some basis vectors \(\i\) and \(\j\)
  2. Where those basis vectors are taken to by the transformation.

How the transformation affects any other point follows from those two pieces of information.

So \(\i\) might be taken to \(a\i + b\j\), and \(\j\) might be taken to \(c\i + d\j\). In this case we would use the following matrix to describe the transformation:

$$ \mat{a}{c} {b}{d} $$

Some examples are

$$ \begin{array}{ll} \text{stretch by a in the i-direction} & \mat{a}{0} {0}{1} \\\\ \text{stretch by a in the i-direction and shear right} & \mat{a}{b} {0}{1} \\\\ \text{rotate anticlockwise 90°} & \mat{0}{-1} {1}{ 0} \end{array} $$

Note that we haven't said what \(\i\) and \(\j\) are yet; they define the 2-dimensional space that we're considering. But, we can think of them for now as the usual orthogonal unit vectors in 2D space.

So the matrix tells us where the basis vectors have been taken to. Any other vector \(f\i + g\j\) is taken to wherever that is using the transformed basis vectors:

$$ f\i + g\j \longrightarrow f\cvec{a}{b} + g\cvec{c}{d} = \cvec{fa + gc}{fb + gd} $$

And that's how matrix multiplication is defined:

$$ \mat{a}{c} {b}{d} \cvec{f}{g} = \cvec{fa + gc}{fb + gd} $$

A matrix represents a linear transformation by showing where the basis vector are taken to.

Change of basis

Suppose person B uses some other basis vectors to describe locations in space. Specifically, in our coordinates, their basis vectors are \(\scvec{2}{1}\) and \(\scvec{-1}{1}\).

When they state a vector, what is it in our coordinates?

If they say \(\scvec{-1}{2}\), what is that in our coordinates?

Well, if they say \(\scvec{1}{0}\), that's \(\scvec{2}{1}\) in our coordinates. And if they say \(\scvec{0}{1}\), that's \(\scvec{-1}{1}\) in our coordinates. So the matrix containing their basis vectors expressed using our coordinate system transforms a point expressed in their coordinate system into one expressed in ours. That last sentence is critical, so hopefully it makes sense! So, the answer is

$$ \mat{2}{-1} {1}{ 1} \cvec{-1}{2} = \cvec{-4}{1}. $$

When we state a vector, what is it in their coordinates?

We give the vector \(\scvec{3}{2}\). What is that in their coordinate system? By definition, the answer is the weights that scales their basis vectors to hit \(\scvec{3}{2}\). So, the solution to

$$ \mat{2}{-1} {1}{1} \cvec{a}{b} = \cvec{3}{2}. $$

Computationally, we can see that we can get the solution by multiplying both sides by the inverse:

$$ \cvec{a}{b} = \mat{2}{-1} {1}{1}^{-1} \cvec{3}{2}. $$

Conceptually, we have

$$ \mat{2}{-1} {1}{1} = \begin{bmatrix}\text{matrix converting their}\\\text{representation to ours} \\ \end{bmatrix} $$

where "their representation" means the vector expressed using their coordinate system. So the role played by the inverse is

$$ \cvec{a}{b} = \begin{bmatrix}\text{matrix converting our}\\\text{representation to theirs} \\ \end{bmatrix} \cvec{3}{2}. $$

When we state a transformation, what is it in their coordinates?

We state a 90° anticlockwise rotation of 2D space:

$$ \mat{0}{-1} {1}{0} $$

what is that transformation in their coordinates? The answer is

$$ \begin{bmatrix}\text{matrix converting our}\\\text{representation to theirs} \\ \end{bmatrix} \mat{0}{-1} {1}{0} \begin{bmatrix}\text{matrix converting their}\\\text{representation to ours} \\ \end{bmatrix} $$

since the composition of those three transformations defines a single transformation that takes in a vector expressed in their coordinate system, converts it to our coordinate system, transforms it as requested, and then converts back to theirs.