Linear Algebra & Complex Numbers Mind Map

Every linear algebra & complex numbers visualization on this site on one map. The branches are the main ideas. Each leaf is a page; the grey line under it is the one thing to remember about it. Click a branch to focus on it, click a page to open it.

Recall mode hides the page names. Read the hint, say the name, then click the "?" to check.

1. How the map is organized

Overview of the linear algebra & complex numbers mind map: Vectors and matrices, 2D transforms, Solving Ax = b, Eigen and factoring, 3D and graphics, Complex and quaternions. — The map at a glance: 6 branches, 29 pages.

Ask what the page does with a matrix and you know its branch:

Does it build the tools? Vectors, products and the determinant are the basics.
Does it move the plane or space? One named move in 2D is a 2D transform; anything with 3D, translation or a camera is 3D and graphics.
Does it find an unknown? Solving for x, the inverse, rank or best fit is solving Ax = b.
Does it break A into simpler matrices? Eigenvalues, QR and SVD are eigen and factoring. Rotation by multiplying numbers is complex and quaternions.

2. The branches side by side

Branch	Key question	Key formula	Watch for
Vectors and matrices	what does A do to î, ĵ?	det = ad − bc	det = 0: space flattens
2D transforms	which move is this?	rotation det 1, reflection det −1	order matters: AB ≠ BA
Solving Ax = b	is there one solution?	[A \| I] → [I \| A⁻¹]	no pivot: none or infinitely many
Eigen and factoring	which directions stay put?	det(A − λI) = 0	not every matrix diagonalizes
3D and graphics	how to place and view a scene?	(x, y, z, 1) homogeneous	divide by w for perspective
Complex and quaternions	how to rotate by multiplying?	e^(iθ) = cos θ + i sin θ	quaternions use half the angle

3. Five facts to remember

The columns of a matrix are where the basis vectors land.
The determinant is the area (or volume) scale factor; a negative one means a flip.
Matrix products apply right to left: in B·A, A happens first.
A square matrix is invertible exactly when its determinant is not 0, that is when it has full rank.
An eigenvector keeps its direction: Av = λv; a real 2D rotation (not 0° or 180°) has none.

See all linear algebra & complex numbers pages in the index, or the other mind maps.