> In the video, it mentions that the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by transforming the unit cube aligned with the basis vectors. It explains that the determinant gives the factor by which volumes are scaled, just as determinants in 2D represent the scaling of areas.
Yes, we all already know that. It is explicitly present as assumed background knowledge in the original question that ivan_ah asked:
>>> How does condensation compute the volume of a parallelepiped from the individual areas of parallelogram?
The question is how the area of the parallelogram defined by the two vectors
[e f]
[h i]
is relevant to the volume of the parallelepiped defined by the three vectors
[a b c]
[d e f]
[g h i]
(as I've phrased it, this question also applies to the textbook way of calculating determinants!)
and the 3b1b video not only doesn't address this, or mention that condensation is a thing you can do, it actively discourages you from calculating determinants by any method.
Yes, we all already know that. It is explicitly present as assumed background knowledge in the original question that ivan_ah asked:
>>> How does condensation compute the volume of a parallelepiped from the individual areas of parallelogram?
The question is how the area of the parallelogram defined by the two vectors
is relevant to the volume of the parallelepiped defined by the three vectors (as I've phrased it, this question also applies to the textbook way of calculating determinants!)and the 3b1b video not only doesn't address this, or mention that condensation is a thing you can do, it actively discourages you from calculating determinants by any method.