SAM · The machine · leaf

Linear Algebra

A matrix is a machine that moves vectors. Linear algebra is learning to read that machine: first by what it does to the grid, then by what it reaches and what it crushes, then by the directions it leaves alone.

Before this leaf: grid coordinates, arithmetic and Pythagoras.
This leaf feeds: Statistics and data, Differentiation.
Objects you will meet again: the 3-4-5 triangle, the two towns, the dial, the fifteen.

building 2 of 20 lessons built

Lessons, notes and demos

One row per lesson, in the order the leaf builds up. Each lesson holds its notes, its demos and its challenges on one page. A demo link opens the lesson at that demo.

Page	Title	Brief	Lesson	Demos	Challenges	Hands forward
01	Vectors	A vector is a size and a direction given together.	Vectors	Walk 5 km Scalar or vector?Same move, anywhere One arrow, both descriptions 3 + 4 is not always 7 The scaling slider Reach the ring	Seed Branch	v = (3, 4) and the 3-4-5 triangle
02	Span	The span is every point you can reach by adding and scaling some arrows.	coming	Paint the spanReach (−2, 5)?Parallel partner	Seed Branch	a second arrow w, and a trail that refused to leave the line
03	Linear independence	An arrow is independent when the others cannot reach it.	coming	Drop one arrow	Seed Branch	a test for redundancy without the word parallel
04	Basis and coordinates	A basis is the smallest set of arrows that reaches everything.	coming	Same arrow, two gridsTwo arrows in 3D	Seed Branch	same object, different numbers
05	Dot product	The dot product is one number that says how much two arrows agree.	coming	Shadow on vPush the car	Seed Branch Canopy	perpendicular means dot product zero
06	Matrices as machines	A matrix is a machine that moves every arrow and keeps the grid straight.	coming	The blendWhere does (3, 4) land?	Seed Branch Canopy	the house matrices S and H
07	Matrix times vector	Ax can be read as a recipe of columns or as dot products with rows.	coming	Two readings of Sv	Seed Branch	the transpose, lightly, with a promise
08	Composition	Two machines one after another make one machine, and the order matters.	coming	Rotate then shearShear then rotate	Seed Branch Canopy	matrix multiplication, derived not memorised
09	Determinant	The determinant is how much the machine stretches area.	coming	Square to parallelogram	Seed Branch Canopy	det = 0 collapses the plane
10	Systems Ax = b	A system asks which input the machine sends to b.	coming	Drag b, column pictureDrag b, row picture	Seed Branch	one, none or infinitely many, and why
11	Inverse	The inverse is the machine that undoes A, when undoing is possible.	coming	Run A then A⁻¹Try to undo P	Seed Branch Canopy	a crushed plane cannot be uncrushed
12	Elimination	Elimination changes the equations without changing the answer.	coming	Step the rows	Seed Branch	pivots count what survives
13	Column space, null space, rank	Rank counts what the machine reaches; the rest it crushes to zero.	coming	Squash space onto a plane	Seed Branch Canopy	eigenvectors live in a null space
14	Projection onto a subspace	A projection is the closest point, and its error is perpendicular.	coming	Drop b onto a plane	Seed Branch	the transpose earns its meaning
15	Least squares	When no exact answer exists, least squares finds the nearest one.	coming	Fit the line	Seed Branch Canopy	regression, for the Statistics leaf
16	Orthonormal bases and Gram–Schmidt	Orthonormal arrows are perpendicular and of length 1, so coordinates become dot products.	coming	Straighten a basis	Seed Branch	the inverse of a rotation is its transpose
17	Eigenvectors	An eigenvector keeps its direction when the machine moves it.	Eigenvectors	The blend Hunt the directions Run the years Matrix lab	Seed Branch Canopy	the two towns and their 0.700
18	Change of basis and diagonalisation	Read the machine in its eigen-basis and it only stretches.	coming	Switch to the eigen-grid	Seed Branch Canopy	powers of A become easy
19	Symmetric matrices	A symmetric machine stretches along perpendicular axes.	coming	Dial to ellipse	Seed Branch Canopy	perpendicular eigenvectors
20	SVD	Every machine is a rotation, a stretch and a rotation.	coming	The shear gives inCompress an image	Seed Branch Canopy	the capstone

Applications that hang from this leaf

Each application hangs from the lessons it needs. It is built once those lessons exist.

Branch	Brief	Needs	Lesson
Homogeneous coordinates	Translation is not a linear machine, so graphics adds one coordinate to make it one.	page 8	coming
Hill cipher	A cipher is a matrix that can be undone mod 26.	page 11; Modular arithmetic leaf	coming
Regression	Least squares on the fifteen is a regression line.	page 15; Statistics leaf	coming
PageRank	A search ranking is the eigenvector of a web graph.	page 18; Graphs and Probability leaves	coming
PCA	The covariance matrix of the fifteen is symmetric, and its eigenvectors point along the data.	pages 19 and 20; Statistics leaf	coming

Use this in your course

Link to any lesson, or to one demo on it: every demo has its own link, shown in the Demos column. You may also copy and adapt these lessons under the CC BY-SA 4.0 licence. Please credit them like this:

සංගතික · Sangathika by Niranga Dharmaratna, CC BY-SA 4.0, https://www.sangathika.com/

More about reuse on the about page.

Glossary

vector page 01: A size and a direction given together; drawn as an arrow, written as a list of components.
span page 02: Every point you can reach by adding and scaling some arrows.
independent page 03: An arrow the others cannot reach; only the zero recipe makes zero.
basis page 04: The smallest set of arrows that reaches everything.
dot product page 05: One number that says how much two arrows agree; zero when they are perpendicular.
matrix page 06: A machine that moves every arrow and keeps the grid straight; its columns say where the basis arrows land.
determinant page 09: How much the machine stretches area; negative means a flip, zero means the plane collapses.
system page 10: The question: which input does the machine send to b?
inverse page 11: The machine that undoes A; it exists only when the determinant is not zero.
elimination page 12: Row moves that change the equations without changing the answer.
rank page 13: The dimension of what the machine reaches.
projection page 14: The closest point in a subspace; its error is perpendicular.
least squares page 15: The nearest reachable answer when no exact one exists.
orthonormal page 16: Perpendicular arrows of length 1.
eigenvector page 17: An arrow that keeps its direction when the machine moves it.
eigenvalue page 17: The number λ that says how much a matrix stretches its eigenvector: Av = λv.
symmetric page 19: A matrix equal to its transpose; it stretches along perpendicular axes.

For the teacher

Sequencing and rationale

Four parts. Part I builds with vectors (span, independence, basis, dot product) so that Part II can read a matrix as a machine described by its columns. Part II runs machine, product, composition, determinant, systems, inverse, elimination, and the four subspaces, in the order need before tool: systems are asked before the inverse answers them. Part III brings length and angle back for projection, least squares and orthonormal bases. Part IV reads the machine on its own axes and ends with SVD, where the shear that defeated eigenvectors gives in. Two house matrices, S and the shear H, recur from page 6 to page 20.