Projections and least squares: find the closest fit
Project a vector onto a direction, measure its orthogonal residual, and connect that geometry to least squares, regression, and nonunique coefficients.
Writing
Clear accounts of what was tried and learned.
Explanations, tutorials, build notes, and essays grounded in work that can be examined. Explore a concept, try an experiment, and follow the reasoning behind it.
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Pseudoinverse: choose the smallest least-squares solution
Understand the Moore–Penrose pseudoinverse through exact and inconsistent systems. Separate residual error from solution norm, inspect projectors, and see how an SVD cutoff changes the problem.
Rotation matrices: turn vectors, track frames, and check the order
Build rotation matrices that preserve length and handedness. Compare fixed-axis rotations in 3D, distinguish rotating a vector from changing its coordinates, and undo a rotation with its transpose.
Singular value decomposition: directions, gains, and low-rank approximation
Build an SVD from orthogonal directions and nonnegative gains. See a circle become an ellipse, identify lost directions, and measure the error from keeping one singular component.
Taylor expansion and linearization: predict locally and check the error
Build constant, linear, and quadratic approximations around a chosen center. Compare their errors, calculate a Taylor remainder bound, and connect the same idea to gradients, Hessians, and Jacobians.
Twists and screw axes: connect point velocities to rigid motion
Build a six-component twist from a screw axis, calculate point velocities, and compare exact helical motion with a tangent prediction. Separate pitch, accumulated displacement, current rate, and pure translation.
Unit quaternions: compose rotations and understand the sign
Rotate vectors with Hamilton quaternions, check composition order, and see why q and minus q describe the same orientation. Includes an interactive experiment and Python.
Bayesian inference: learning a robot's grasp success rate
Learn Bayesian inference through a robot grasp example. Update a Beta prior with successes and failures, then distinguish uncertainty from prediction.
Conditional probability: reading a robot sensor
Learn conditional probability with an interactive robot sensor example. Work through the formula, Bayes' rule, base rates, and common mistakes using counts.
Cross product: find a normal direction and calculate torque
Calculate a three-dimensional cross product, follow the right-hand rule, and connect its magnitude to area. Explore signed torque with a movable lever arm and force.
Determinants: signed area, volume, and collapsed directions
Calculate a determinant, see how its sign records orientation, and connect zero area to singular matrices. Learn why a small determinant alone does not imply poor conditioning.
The dot product: angles, projections, and robot motion
Learn the dot product with an interactive vector diagram, a worked projection example, and a robot heading calculation. Includes exercises and solutions.