Linear transformations as matrices. Need better explanation

Hello! Need help with understanding what linear transformation means and how it works.
For matrix:
[[3, 1],
[1, 2]]

This is the excerpt from the video:

The point 0, 0 gets sent to 0, 0 which always happens. 
The point 1 0 gets sent to 3, -1 which is the bottom corner. 
The point 0 1 gets sent 2, -3 and the point 1 1 gets sent to 5 2.

There is no explanation in the video other than this. Just a fact that The point 1 0 gets sent to 3, -1 I don’t understand why point 1, 0 gets sent to 3, -1 why not to 25, -32 or to 19, 27? Could anyone help and explain please? This particular video seems quite useless for understanding

Hi @burrito can you tell me which one is that video?

it is Linear transformations as matrices video under Linear transformations section

Ok. To understand why the point (1, 0) gets sent to (3, -1) under the linear transformation defined by the matrix

[[3, 1],

It’s important to understand how matrix multiplication works in the context of linear transformations.

A linear transformation in a two-dimensional space can be represented by a 2x2 matrix. When we apply this transformation to a point, we multiply the matrix by a column vector representing the point.

To find where the point (1, 0) is sent, we multiply this matrix by the column vector representing (1, 0).

The multiplication looks like this:


So, the point (1, 0) under this transformation is sent to the point (3, 1), not (3, -1) as mentioned in your excerpt. There may be a typo or misunderstanding in the video’s explanation.

After applying the linear transformation defined by your matrix, here are the transformed points:

  • The point (0, 0) gets sent to (0, 0). This is true for any linear transformation as the origin point remains unchanged.
  • The point (1, 0) gets sent to (3, 1)
  • The point (0, 1) gets sent to (1, 2).
  • The point (1, 1) gets sent to (4, 3).

These transformations are the result of multiplying the matrix with each point represented as a column vector.

I think the discrepancy can be because in the video there are difference with the default settings of the lab.

I hope this helps!

Thank you @pastorsoto ! now it is much clearer