Day 4

The full equation.

We've found the particular solutions that's needed to create the general solution!

For clarity, let's discuss what we did:

1. Find the particular solution to Ax = B
2. Find all the solutions to Ax = 0
3. Add steps one and two together

With these steps finished, congratulations! you've become familiar with particular and general solutions. From here on out, you know that there are cases where large unknown variables with little equations can be solved with the right conditions.

The general solution

$$\left\{ x \in \mathbb{R}^5 \mid x = \begin{pmatrix} 42 \\\ 8 \\\ 0 \\\ 12 \\\ 0 \end{pmatrix} + \lambda_1 \begin{pmatrix} 8 \\\ 2 \\\ -1 \\\ 4 \\\ 0 \end{pmatrix} + \lambda_2 \begin{pmatrix} -4 \\\ -12 \\\ 0 \\\ 7 \\\ -1 \end{pmatrix}, \; \lambda_1, \lambda_2 \in \mathbb{R} \right\}$$

P.S. Hope you got the answer right!

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: 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)