Linear Algebra

Vectors, matrices, eigenstuff — the language of robotics.

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Why it matters in robotics

Linear algebra is the language of robotics: every rigid-body pose, rotation, camera projection, and state estimate is a vector or matrix operation. Rotation and transformation matrices are what place objects and sensors in space and describe how they move, and the same operations are what let a robot combine noisy measurements into a single reliable estimate, solve for the best-fitting answer when it has more data than unknowns, and know when a system of equations has no solution, one solution, or many. It is the foundation that kinematics, control, estimation, and SLAM are all built on.

What to study

✓Vectors, matrix multiplication, and linear transformations as geometric operations (rotation, scaling, projection)
✓The four fundamental subspaces: column space, null space, rank, and what they imply about Ax = b solvability
✓Eigenvalues, eigenvectors, and diagonalization — the basis of stability analysis and principal directions
✓Orthogonality, least-squares, and the SVD for solving overdetermined systems and low-rank approximation

Where to study

Cheatsheet

Symbols used below

A

m \times n

matrix (a linear map)

x, b

vectors in

A x = b

\mathrm{rank}(A)

dimension of the column space

N(A)

null space:

\{x : Ax = 0\}

\lambda, v

eigenvalue, eigenvector

U, \Sigma, V

SVD factors:

A = U \Sigma V^\top

\sigma_i

singular value (diagonal entry of

\Sigma

)

Solvability of $Ax = b$

A solution exists iff $b$ is in the column space of $A$ .

If it exists, it's unique iff $N(A) = \{0\}$ (columns independent). Otherwise solutions form a family: one particular solution plus anything in $N(A)$ .

Rank-nullity

$\mathrm{rank}(A) + \dim N(A) = n$ (number of columns).

More independent columns leaves less room for a null space — full column rank forces $N(A) = \{0\}$ .

Eigenvalues & eigenvectors

$Av = \lambda v$ — the transform only scales $v$ , it doesn't rotate it off its own line.

If $A$ has $n$ independent eigenvectors, $A = P \Lambda P^{-1}$ with $\Lambda$ diagonal — repeated matrix powers and stability both reduce to looking at the $\lambda_i$ .

SVD

$A = U \Sigma V^\top$ for any matrix (square, rectangular, singular — always exists).

$V^\top$ rotates, $\Sigma$ stretches along orthogonal axes by $\sigma_i \ge 0$ , $U$ rotates again. Keeping only the largest $\sigma_i$ gives the best low-rank approximation of $A$ .

Least squares

When $Ax = b$ is overdetermined (more rows than columns, no exact solution), the best-fit $\hat{x}$ minimizes $\|Ax - b\|^2$ .

It solves the normal equations: $A^\top A \hat{x} = A^\top b$ .

An eigenvector's direction is unchanged by the transform — it is only scaled by its eigenvalue λ. A general vector gets rotated off its original line.

	Eigendecomposition	SVD
Applies to	Square matrices only	Any $m \times n$ matrix
Requires	$n$ independent eigenvectors to exist	Always exists
Axes used	Eigenvectors (not generally orthogonal)	Orthonormal input/output bases $U$ , $V$
Typical use	Stability, repeated dynamics ( $A^k$ )	Rank, pseudo-inverse, low-rank approximation

The SVD A = UΣVᵀ maps the unit sphere to an ellipsoid: Vᵀ rotates, Σ scales each axis by a singular value σᵢ ≥ 0, and U rotates again. The singular values are those axis lengths.

Application focus

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