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41 lessons found

Foundations first

  1. Foundations / 9 min read

    Vector spaces: build direction from addition and scaling

    Learn vector spaces through robot displacement. Explore span, linear independence, basis, and dimension with an interactive diagram, Python, and exercises.

  2. Foundations / 9 min read

    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.

  3. Foundations / 9 min read

    Eigenvalues and eigenvectors: find the lines a matrix preserves

    Understand eigenvalues through 2D transformations. Test stretching, reversal, zero eigenvalues, rotation, and repeated values, then connect them to robot error dynamics.

  4. Foundations / 10 min read

    Tensors: read shapes, select values, and move axes

    Learn tensors through a robot image batch. Explore shape, indexing, slicing, and axis order, then compare reshape with transpose using a runnable Python example.

  5. Foundations / 9 min read

    Vector norms and normalization: L1, L2, and L∞

    Measure vectors with L1, L2, and infinity norms. Explore unit boundaries, normalize a robot displacement, and distinguish zero vectors from tiny nonzero inputs.

  6. Foundations / 10 min read

    Matrix multiplication: calculate entries and compose transformations

    Learn matrix multiplication through row-column dot products, compatible shapes, and a rotation-and-stretch experiment that shows why transformation order matters.

  7. Foundations / 11 min read

    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.

  8. Foundations / 9 min read

    Matrix transpose and inverse: when do they agree?

    Transpose rectangular matrices, calculate a 2×2 inverse, and test when a transpose reverses a transformation. Explore rotations, reflections, and singular maps.

  9. Foundations / 11 min read

    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.

  10. Foundations / 12 min read

    Rank and null space: reachable outputs and hidden input changes

    Use rank, column space, and null space to understand a linear map. Explore rank-nullity, unreachable targets, and families of solutions with a small matrix experiment.

  11. Foundations / 10 min read

    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.

  12. Foundations / 12 min read

    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.