How to find the transformation matrix of a linear transformation

The transformation matrix is a representation of the transformed standard basis
vectors. For example, in a 2-dimensional coordinate system if the transformed coordinates of the unit vector are,

and that of unit vector are,

These two basis vectors can be combined in a matrix form,

M is then called the transformation matrix.

Also, any vector can be represented as a linear combination of the standard
basis vectors. For example, if is a 3-dimensional vector such that,

then can be described as the linear combination of the standard basis vectors,

This property can be extended to any vector.

Now, let’s say we want to determine the transformed coordinates of the vector,

Multiplying the vector with the transformed basis vector matrix,


So in general any vector can be transformed by multiplying it
with the transformation matrix
The general form for transformation can be derived as,


Hence, is a the general form of the transformation matrix.
Any vector which is passed into this matrix will be transformed. Why not try passing the 2-dimensional standard basis vectors into this and see what comes out?

The unit vector along the x-axis will be transformed to,

The unit vector along the y-axis will be transformed to,

If we now combine the transformed unit vectors and into a matrix,

And we get our original transformation matrix back.