Basis of a Vector Space | Linear Algebra

Abide by Reason clarifies the notion of a basis for students who have met linear combinations and span but have not yet internalized what makes some lists of vectors “just right” for describing an entire space.

Abide by Reason revisits span for learners who know how to form linear combinations but have not fully connected span with the idea that a small set of vectors can generate an entire space.

Abide by Reason sharpens the concept of linear independence for students who have heard the term but need a precise algebraic test tied to an equation.

Abide by Reason grounds the abstract definition of basis for students by returning to the familiar coordinate plane (\mathbb{R}^2).

Abide by Reason introduces dimension for learners who can recognize bases in simple spaces but have not yet seen how basis length becomes an invariant measuring “degrees of freedom.”