I am having some problem with understanding linear transformation along with some questions.

If someone can check my understanding is correct or not:

In the Matrices as linear transformations video, we have matrix [[3,1],[1,2]]. And linear transformation means the original plane like [1,0],[0,1], after multiply with this matrix, we got a final plane in the right.

I am not getting this part, the chatGPT is saying original plane e1= [1,0]
after T(e1) = (1,0) still, how? why the value needs to be the same as the e1 before transformation?

To answer your first question:
For linear transformations, the transformation matrix M must be multiplied with the vector v in the order M * v, as a rule in linear algebra. Multiplying it in the order v * M would not apply the linear transformation correctly.

To answer your second question:
It is possible to end up with the same vector when applying some linear transformations. In the example you gave, it just so happens that applying that shear transformation to the vector e1 results in the same vector.

In this particular case, when you calculate the linear transformation T(e1), you get T([1, 0]) = [1x1 + mx0, 0x1 + 1x0] = [1, 0], which happens to be e1 again.

This does not apply for every linear transformation, however. Take for instance this shear transformation matrix:

\left(\begin{array}{cc}
1 & 0\\
m & 1
\end{array}\right)

When you apply this shear transformation to e1, you’ll end up with [1x1 + 0x0, 1xm + 1x0] = [1, m], which is no longer e1.

To answer the first question:
Say for example you have the transformation matrix M of the general form:

\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)

and vector v:

\left(\begin{array}{cc}
x\\
y
\end{array}\right)

If you do M * v, you would get:

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

If you do v * M (after reshaping v so matrix multiplication is valid), you would get:

\left(\begin{array}{cc}
x & y\\
\end{array}\right)
\left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)
=
\left(\begin{array}{cc}
ax + cy & bx + dy
\end{array}\right)

So these would not be equal. In your example matrix, it seems to have given the same result since b and c were both equal to 2, but that is not always true.

For your second question, I think that is incorrect because the value of the vector v[0] and v[1] can change depending on what you set v to be. The values in the transformation matrix needs to be constant, and must not depend on the v itself.