Linear algebra: directions, systems and approximation
Use vectors and matrices to model constraints, diagnose solution structure, fit inconsistent observations and interpret preserved directions.
Choose any lesson or try the project first. Shared lessons retain the same saved work wherever you enter them. Completing a lesson records participation; practice, transfer checks and independent reasoning remain separate.
Recommended preparation and ways to catch up
- Read coordinate pairs and substitute into linear expressions.
- Matrix shapes, dot products and symbols are introduced in the shared lessons.
Find a foundation course · Open contextual refreshers. Each investigation includes a short recap; preparation never locks access.
Loading local course progress…
Your units and lessons
Choose a unit to see its lesson sequence. Every lesson opens on its own page.
Coordinates and linear maps
Distinguish coordinates, span and matrix action.
Build a point from directions
Can two selected vectors reach every point in the plane?
Trace vectors through a matrix
Compute dot products, check shapes and compose transformations in order.
Turn a production plan into resource needs
How can one table calculate several resource totals?
Exact and approximate solutions
Use rank and residuals to distinguish solvability from best approximation.
When do equations identify one point?
How can row subtraction reveal one, no or infinitely many solutions?
Fit a line when equations disagree
Which line minimizes the squared residuals of three observations?
Projection and preserved directions
Explain information loss, eigenvectors and repeated transformations.
Project a cloud and inspect what is lost
Connect unit directions, reconstruction, covariance, eigenvectors and low-rank intuition.
Find a direction a matrix preserves
When does applying a matrix only scale a selected nonzero vector?
Model and audit a small linear system
Choose a resource, measurement or transformation scenario. Submit a matrix with units, an exact-system diagnosis, a least-squares comparison and an interpretation of a projection or eigen-direction. Keep exact equations, approximate fits and decisions separate.
Build from lesson evidence, compare alternatives, write an artifact, and tackle unfamiliar numerical cases. Use the rubric with a teacher, peer or colleague to assess the quality of your reasoning.
Open the course project