Question regarding matrix linear transformation

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

If someone can check my understanding is correct or not:

  1. 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.

my first question, why we use matrix to multiply the vector M * v not v * M?

  1. in the week3 assignment

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?

Hi @Bio_J!

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.

I hope this helps clear things up!


Here comes with another 2 questions then:

What do you mean by

Can you explain why it’s incorrect?
Like in the following M * v and v * M the result is same:

  1. In the assignments it didn’t say after transformation e1 must be = [1,0] so why can’t I just apply

Hi @Bio_J,

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.

I hope this helps!