Linear Algebra
Transforming the plane…
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Linear Algebra
Transforming the plane…
Matrices are not tables of numbers — they move space. Build the intuition in 2D, where you can drag everything, then step into 3D only where the idea needs a third dimension.
Scale two vectors and add them. Every point you can reach is their span — the whole plane, or a single line when the vectors line up.
c₁a + c₂bVector Addition and ScalingMix and ReachA matrix moves the whole plane, and its columns say exactly where î and ĵ land. Drag them and watch the grid follow.
Mv = v₁ (column 1) + v₂ (column 2)Linear Combinations and SpanColumns Are ImagesRow times column for the numbers, and one transformation after another for the picture — which is why AB and BA differ.
(AB)ᵢⱼ = row i of A · column j of BMatrix as a TransformationCompose the MapsThe factor by which a matrix scales area in 2D and volume in 3D. Negative means flipped; zero means squashed flat.
det A = ad − bcMatrix as a TransformationArea Scale FactorRow operations reduce a 3×3 system step by step, while three planes in 3D keep meeting at the same point — or never meet, or share a line.
[A | b] → row echelon form → RREFLinear Combinations and SpanRow OperationsRow-reduce [A | I] until the left side is I and the right side is A⁻¹, then watch A⁻¹ put the plane back where it was.
[A | I] → [I | A⁻¹]Gaussian EliminationUndo the MapA matrix that squashes 3D space onto a plane or a line, the vectors it sends to zero, and why the two always add up to three.
rank + nullity = nGaussian EliminationWhat SurvivesSweep a vector around the circle and light up the ones that stay on their own line. The eigenvalue is how much they stretch.
Av = λvDeterminantStay on Your LineOne vector, two coordinate grids. The basis matrix converts one way and its inverse converts back.
v = B[v]_B, [v]_B = B⁻¹vInverse MatricesTwo GridsFit a line to points you can drag. The best fit is a projection onto the column space, and the residuals stand perpendicular to it.
AᵀA x̂ = AᵀbVector ProjectionProject to FitMultiply 2×2 blocks with seven products instead of eight, and recurse — the first algorithm to beat n³.
T(n) = 7T(n/2) + O(n²)Matrix MultiplicationDivide and ConquerMatrix products are associative but not equally cheap. Dynamic programming finds the bracketing that does the least work.
m[i][j] = min over k of m[i][k] + m[k+1][j] + pᵢ₋₁pₖpⱼMatrix MultiplicationInterval DPMost people meet linear algebra as rules for shuffling numbers. Seen as the plane bending — columns as where the axes land — the rules stop being arbitrary and start being obvious.
A neural network layer is a matrix, training is least squares in disguise, and PCA is an eigenvector problem. Every model you run is linear algebra at scale.
Planes meeting at a point, space squashed onto a line, a box's volume as a determinant — these are hard to imagine and easy to see. Orbit the scenes until they click.