Linear transformations as matrices. Need better explanation

Course Q&A Math for Machine Learning Linear Algebra for Machine Learning and Data Sc...
1

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

2

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

3

it is Linear transformations as matrices video under Linear transformations section

4

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

[[3, 1],
[1,2]]

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:

matrix-multiplication2

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!

5

Thank you @pastorsoto ! now it is much clearer