Determinants and Volume: A Geometrical Explanation of Determinants (Determinants Done Right Part 1)
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- Опубліковано 28 жов 2021
- An exploration of determinants and their relationship to n-dimensional oriented volume, including:
- History of determinants
- Geometrical motivation and intuition for determinants
- Fundamental algebraic properties of determinants
- Explanation of Leibniz and Laplace expansions
More videos on determinants: • Determinants
man i watched your videos so many time, awesome content, really thanks
this is so big brain. only observing some basic (and super) identities, then starting from there.
How is this so underrated! You will eventually blow up, and I will rejoice as once again, I was watching blargoner before it was cool.
I'm loving this channel so far but someone's got to explain 10:16 to me.
I can't see how I should prove:
A(x + y, z) = A(x, z) + A(y, z)
from the provided properties:
αA(x, y) = A(αx, y) = A(x, αy)
A(x, y) = A(x, x+y) = A(x+y, y)
Geometrically, it's clear to me.
If z = 0, show that both sides are zero using the first property. If z != 0, extend to a basis w,z of the plane. Now if x = aw + bz, show that A(x,z) = aA(w,z) by first considering the case b = 0 and then the case b != 0. Similarly if y = cw + dz, show that A(y,z) = cA(w,z). Finally show that A(x+y,z)=(a+c)A(w,z).
@@blargoner Ohhh, thanks! I didn't think to do the substitution.
12:20 well, you’d need a metric to know what a “standard basis” is. To say that the area is one, you’d need to have at least the norms of the vectors to say that they’re reciprocals of each other.
You don't need a metric to say that the area is one if you're making it so by definition by choosing an area function that assigns the value 1. It's analogous to drawing a mark on a blank straightedge to determine the unit of measurement.