Educational Cards
Learn from video content, text, and interactive tasks
Filters
Bridge toward Cramer and signed volumes
Understanding how linear maps move oriented normals prepares determinant-based solution formulas in...
Linear system as unknown combination of columns
The system Amathbfx=mathbfb asks which linear combination of columns of A produces mathbfb. Write...
2×2 picture: area ratios
In mathbbR^2, columns mathbfa_1,mathbfa_2 span a parallelogram. Replacing mathbfa_1 by mathbfb...
3×3 parallelepiped story
In mathbbR^3, columns of A span a parallelepiped. Interpret mathbfb inside that solid; ratios of...
Algebraic cofactor mirror
Minors and cofactors reorganize the same volume ratios into Laplace expansion form [@2]. The...
Failure modes and teaching payoff
Near-zero denominators amplify noise; singular systems need consistency checks, not blind division...
Transition to change of basis formulas
Coordinate remapping reinterprets the same geometry in new language. Chapter 13 develops change of...
Mean, square-root mnemonic
If two numbers share mean m and product p, they sit symmetrically about m: write them as mpm d for...
Neurons: weighted sums, bias, then a nonlinearity
This chapter poses a core design question: given activations in one layer, what knobs let a network...
Why depth needs nonlinear breaks
Before the algebra, the narrative asks for a hope about hidden layers: maybe a penultimate layer...
MNIST-style digits as a running example
The running example is handwritten digit recognition on 28times 28 grayscale images, 784 pixels...
Parameter counting and width vs depth intuition
Every weight and bias is a learnable knob. The demo network connects all 784 inputs to each of 16...