LA 101
M12 · L01
Module 12: Capstone

Putting It All Together

Every idea in this course asks one question: how does a matrix act on a vector? Vectors, transformations, decompositions, eigenvalues — all are answers to that question from different angles.

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LA 101
M12 · L01
Four Fundamental Subspaces

The Big Picture

Every m × n matrix defines four subspaces that describe its action completely: column space, null space, row space, and left null space. Rank + nullity = n — always.

Rank-Nullity
r + (n - r) = n
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LA 101
M12 · L01
The Chain of Ideas

Vectors → Matrices → Maps

Vectors are points. Matrices are functions. Matrix multiplication is function composition. The identity matrix is the identity function. This single conceptual leap unifies the entire course.

v
Vector
A
Linear map
Av
Action
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LA 101
M12 · L01
Ax = b → Least Squares

Solvability and Projection

Ax = b is solvable iff b ∈ C(A). When b is outside the column space, project b onto C(A) — that is least squares. The normal equations and QR factorization compute this projection efficiently.

Normal Equations
AᵀAx̂ = Aᵀb · solution: x̂ = (AᵀA)⁻¹Aᵀb
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LA 101
M12 · L01
SVD — The Universal Decomposition

Rotation · Stretch · Rotation

Any matrix decomposes as A = UΣVᵀ. U and V are orthogonal rotations; Σ is a diagonal stretch. SVD makes everything visible: subspaces, rank, condition number, best low-rank approximation.

SVD
A = U\Sigma V^T = \sum_{i=1}^{r}\sigma_i u_i v_i^T
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LA 101
M12 · L01
Choosing a Decomposition

Right Tool, Right Problem

  • LU: Solve Ax = b once efficiently
  • QR: Least squares — more stable than normal equations
  • Eigendecomposition: Matrix powers, stability, dynamics
  • SVD: Low-rank approx, pseudoinverse, PCA
  • Cholesky: Symmetric positive definite — half LU cost
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LA 101
M12 · L01
Exploit Structure

Symmetry · Sparsity · Toeplitz

Real matrices are rarely general. Symmetric matrices guarantee orthogonal eigenvectors. Sparse matrices enable iterative solvers. Toeplitz (convolution) is diagonalized by DFT — FFT gives O(n log n) matrix products.

O(n²)
Dense MV
O(n log n)
FFT (Toeplitz)
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LA 101
M12 · L01
Linear Algebra Everywhere

One Math, Many Disciplines

The same ideas appear under different names: convolution → Toeplitz, Fourier → orthonormal basis, PCA → spectral decomposition, neural networks → matrix compositions, quantum states → Hilbert space vectors.

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LA 101
M12 · L01
Think Geometrically

Draw It Before You Compute

Before computing, picture the geometry: projection onto a subspace, stretch factor of a map, angle between vectors. The condition number κ₂(A) = σ₁/σₗ, with l = min(m, n), tells you how many digits of accuracy you will lose — geometry predicts numerical disaster before it strikes.

Condition Number
κ₂(A) = σ₁ / σₗ · lose ≈ log₁₀(κ₂) digits
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LA 101
Knowledge Check

Check whatstuck

Four questions synthesising the course — solvability, decompositions, structure, and conditioning.

Question 1 of 0
Score 0/0

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LA 101
Key Takeaways
Summary

Key Takeaways

  • One question: how does a matrix act on a vector?
  • Four subspaces describe that action completely; rank + nullity = n
  • SVD A = UΣVᵀ is the universal decomposition — use it by default
  • Choose the right decomposition for the task at hand
  • Exploit structure: symmetry, sparsity, Toeplitz speed everything up
  • Think geometrically before computing algebraically
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LA 101
Capstone
Module 12 · Lesson 1 Complete

The Web Is Complete

You have traversed the full arc of linear algebra — from geometric vectors to SVD, from Gaussian elimination to neural networks. Every thread leads back to Ax. That is the beauty of the subject.

Continue
M12-L2: Advanced Topics →
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