Vectors.
Two moves, scale and add. Everything else follows.
Predict first
Average two images, pixel by pixel: a handwritten 3 and a handwritten 8. What comes out?
Span
Two arrows. Every place they can reach.
Turn c and d. Every arrow c v + d w is a linear combination, and all of them together are the span.
Drag the tips of v and w
Trapped
Reach the star. Then find where you can’t.
Use only c v + d w. Line the arrows up, or step into space, and watch the star slip out of reach.
The ghost
Halfway between a 3 and an 8. Both, and neither.
Two real digits are two points, 784 numbers each. Mix them pixel by pixel, and exactly halfway is a ghost.
A broken rule
Same numbers, another order. Another answer.
On paper, (a + b) + c is always a + (b + c). Every vector sum leans on that rule. Your computer only nearly keeps it.
(a + b) + c =
a + (b + c) =
When you type , your computer stores exactly
Random triples of one-decimal numbers, 0.1 to 0.9.
Basis
Reach everything. Waste nothing.
Drag the rulers b₁ and b₂, or the point p. The point never moves. Only its numbers change.
Drag b₁, b₂ or p
Standard rulers:
Your rulers:
Walk the line
From one digit to another. In a straight line.
Every step is (1 − t) A + t B. Walk it, and see where each step lands in the space a network learned.
What the nearest digits say, all along the line
Drag to turn
Straight among pixels. Bent in the space that understands digits.
Digits from MNIST, by Yann LeCun, Corinna Cortes and Christopher J. C. Burges (CC BY-SA 3.0).
Prove it
Four questions. The reasons matter more than the score.
Question 1 of 4
v = (3, 2). What is −2 v?
Question 2 of 4
Two arrows in space, neither a multiple of the other. What is their span?
Question 3 of 4
Can three arrows be a basis for the plane?
Question 4 of 4
Why can (0.1 + 0.2) + 0.3 differ from 0.1 + (0.2 + 0.3) on a computer?