Day 16

A question.

I haven't found the answer and probably never will considering it took me way longer than I imagined. So I'll just leave it here :D

The equation for finding transformation matrix.

$$\Phi (b_k) = \Sigma_{i = 1}^4 \alpha_{ik} c_i \text{ for } k = 1, \dots, 3$$

The solution I thought and that haven't been disproven (a.k.a not scientifically proven)

Just... transpose it.

Here's the formula

$$\Phi (b_1) = C_1 - C_2 + 3C_3 - C_4 \\\ \Phi (b_2) = 2C_1 + C_2 + 7C_3 + 2C_4 \\\ \Phi (b_3) = 3C_2 + 1C_3 + 4C_4$$

Split it and remove the constants

$$\left[\begin{array}{cccc} 1 & -1 & 3 & -1 \\\ 2 & 1 & 7 & 2 \\\ 0 & 3 & 1 & 4 \end{array}\right]$$

Notice that with the result before, each row corresponds to a column?

$$\left[\begin{array}{cccc} 1 & -1 & 3 & -1 \\\ 2 & 1 & 7 & 2 \\\ 0 & 3 & 1 & 4 \end{array}\right]^T = \left[\begin{array}{ccc} 1 & 2 & 0 \\\ -1 & 1 & 3 \\\ 3 & 7 & 1 \\\ 1 & 2 & 4 \\\ \end{array}\right]$$

So change the row into columns and vice versa!

$$\left[\begin{array}{ccc} 1 & 2 & 0 \\\ -1 & 1 & 3 \\\ 3 & 7 & 1 \\\ 1 & 2 & 4 \\\ \end{array}\right]$$

Seriously, what's math?? What am I doing wrong? I know the guy from Cambridge who wrote this is much smarter than I am so he must have thought someone will think this is just a transposed version. So why :(

Acknowledgement

I can't overstate this: I'm truly grateful for this book being open-sourced for everyone. Many people will be able to learn and understand machine learning on a fundamental level. Whether changing careers, demystifying AI, or just learning in general, this book offers immense value even for fledgling composer such as myself. So, Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong, thank you for this book.

Source: Axler, Sheldon. 2015. Linear Algebra Done Right. Springer Deisenroth, M. P., Faisal, A. A., & Ong, C. S. (2020). Mathematics for Machine Learning. Cambridge: Cambridge University Press. https://mml-book.com (opens in a new tab)