What is...a projective space?

Goal.
Explaining basic concepts of linear algebra in an intuitive way.

This time.
What is...a projective space? Or: My lines cross.

This is the last video on this playlist. Thank you for watching, and for taking this journey with me. I enjoyed it very much, and I hope you had fun as well.

Slides.

Mathematica.

#linearalgebra
#algebra
#mathematics

[solveig]

I am doing a presentation on projective geometry and tried to find the Wolfram alpha demonstration, but it is not there. How did you find the one you used? Your video was great

[mattiapiras]

Nice video! if I may Sir, I'd have a question, and I apologize in advance if it is trivial or stupid. This concept can be, as you mentioned, generalized in a multivariable system. If this starting nD system of figures, or for better saying, of equations, is projected in a greater dimensional space (that could be, as analogy, an hypersphere centered in the origin with radius=1) I will end up with an (n+1)D system with a new variable, let's call that 'h', and an additional equation representing the previously mentioned hypersphere (x1² + ... + xn² + h² -1 = 0). Correct me if I am wrong, so, homogenizing the system with the variable h means multiplying each monomial of every equation by h as many time as needed to have the same degree for all terms in the same equation but every equation can have different degree from the other. If I found a solution at infinity and if I computed the Jacobian of the new homogenized system in that solution (so original system but homogenized plus the equation of the hypersphere) this would make it rank deficient, if I am not wrong. So, this solution at infinity seems to be singular but I can't understand the real motivation. Do you have any thoughts about it, Sir? Do you think that it is caused by the homogenization process or is because I'm thinking inside R^n? Sorry if this concept it is not expressed in a very formal way, but after having seen your video I tried to think about it more in detail but I'm a little stuck since I'm not an expert in mathematics :) . Thanks in advance

[cherma]

So, from your video I got the intuition, that I can represenr a higher dimensional space by it's projective space. Meaning R^3 can be represented by RP^2 - why? Because I have a topology on RP^2 that preserves the information of R^3 via defined mapping in RP^2 for example Vector addition. Therefore the analogy of the painting fits so well because it maps RP^2 to R^3. Meaning if I would add(?) one more cell complex to RP^2 it would allow me to switch back to R^3. 🤔

Thank you for the informative video, greetings from Germany! 👋

[samirelzein]

tried your best, you need to prepare more, write your script if you want. Good job, keep them coming.