Linear Algebra
See space move — then name what you saw with matrices, determinants, and eigenvectors
1. Why this matters
Linear algebra is the language of anything that stretches, rotates, mixes, or projects space: computer graphics, data science, quantum mechanics, circuits, and finite-element engineering. The usual wall is that classes jump straight to row operations. This lesson does the opposite — you watch what a matrix does to the plane, then learn the names for what you already saw.
If you remember one picture from this page, make it this: a matrix is not a grid of numbers. It is a machine that takes every vector and moves it somewhere else — linearly.
2. Vectors as arrows
A vector is an arrow: a direction and a length. In 2D we write it as a column meaning “go x along the horizontal, y along the vertical.”
The special unit arrows along the axes are the basis vectors and . Every other vector is a recipe: “this much î plus that much ĵ.”
Where does the tip of land if you start at the origin? Sketch it mentally before revealing.
3. Matrices as transformations
Apply a matrix A to every point on a grid and the grid deforms — but gridlines stay parallel and evenly spaced. That is what “linear” looks like. Scrub the story below; pause on shear vs reflection and notice what happens to î (red) and ĵ (green).
The 3Blue1Brown punchline: the columns of A are simply where î and ĵ land. Everything else follows from linearity — a vector lands at .
Worked example: 2D rotation matrix
Rotate every point by angle θ counterclockwise. Where does î = [1, 0] land? At [cos θ, sin θ]. Where does ĵ = [0, 1] land? At [−sin θ, cos θ]. Those become the columns:
Check: det(R) = cos²θ + sin²θ = 1 — rotation preserves area. Also R⁻¹ = Rᵀ = R(−θ), so it is an orthogonal matrix (lengths and angles preserved).
For the shear matrix , where do î and ĵ land? What does that tell you the columns are?
4. Determinant as area
The determinant answers one geometric question: how does this transformation scale areas (in 2D) or volumes (in 3D)? Scrub the parallelogram story — the columns of A become the sides of that shape.
If det = 0, the parallelogram collapses to a line (or a point) — the matrix is singular and loses a dimension. Negative determinant means orientation flipped (like a reflection).
Worked example: compute det and read the geometry
For , det(A) = 2·3 − 1·0 = 6. Areas grow by a factor of 6.
Columns: [2, 0] and [1, 3]. The unit square stretches into a parallelogram whose base is length 2 and height is 3 — area 6. No flip (det > 0).
Does the reflection matrix have a positive or negative determinant? What does that mean geometrically?
5. Eigenvectors & eigenvalues
Most vectors change direction when A hits them. Eigenvectors are the stubborn ones: they stay on their line and only get stretched (or flipped). The stretch factor is the eigenvalue λ.
Worked example: eigenvalues of A = [[2, 1], [1, 2]]
For λ₁ = 3: (A − 3I)v = 0 gives v₁ = [1, 1]ᵀ. For λ₂ = 1: v₂ = [1, −1]ᵀ.
Geometric meaning: A stretches space by 3 along the diagonal [1, 1] and leaves the anti-diagonal [1, −1] unchanged. That is exactly what the animation showed.
For the same A = [[2, 1], [1, 2]], is [1, 0] an eigenvector? Check by computing Av and seeing if it is a scalar multiple of [1, 0].
6. SVD — the deeper geometry
Singular Value Decomposition says every matrix (even non-square) is three simple moves: rotate, stretch along axes, rotate again. Scrub the panels left to right.
Worked example: SVD and image compression
An m×n image can be treated as a matrix A. Keep only the top k singular values: .
- Vᵀ — rotate input space onto the “natural axes”
- Σ — stretch by singular values (importance of each axis)
- U — rotate into output (pixel) space
Storage drops from m·n to about k(m + n + 1). For a 1000×1000 image with k = 50, that is roughly 95% savings with often little visible loss — because small singular values carry fine detail you can discard.
7. Practice & applications
The same geometric ideas power ranking, dimension reduction, and graphics. Use these to check that the pictures still guide the algebra.
Application sketch: PageRank as an eigenvector
Model the web as a matrix M whose columns describe “probability of following a link.” The PageRank vector r satisfies Mr = r (eigenvalue 1) — a steady-state distribution of attention.
Geometric meaning: after the link-following transformation, the importance vector does not change direction in probability-space. The dominant eigenvector is the ranking.
Application sketch: PCA
Principal Component Analysis finds eigenvectors of the covariance matrix. The top eigenvector points along the direction of greatest variance in the data cloud — the “longest axis” of the blob.
Same instinct as before: eigenvectors are the natural axes of a linear action (here, of covariance).
A transformation doubles every length and preserves orientation. What is its determinant in 2D? In 3D?
True or false: if Av is parallel to v for a nonzero v, then v is an eigenvector of A.
8. Resources
Go deeper with the classic visual series and a rigorous course. Come back and re-scrub the stories when a formula feels abstract again.