How are standard Cartesian coordinates reinterpreted as scalars acting on basis vectors in linear algebra?

To form a basis, a set of vectors must be both linearly independent and span the target space. Linear independence means each vector contributes a brand-new dimension that cannot be recreated by the others.

Conditions: The vectors belong to the target vector space; The set is minimal (no redundant vectors)

When a third vector pokes out of the plane at an angle, scaling it allows the entire 2D sheet to slide up and down through space. With three freely varying scalars ($a\vec{v} + b\vec{w} + c\vec{u}$), the span now encompasses every conceivable point in the 3D volume.

Conditions: The first two vectors span a plane in 3D space.; The third vector is not on that plane (it pokes out at an angle).; The scalars $a$, $b$, and $c$ vary freely.

A basis is a set of linearly independent vectors that span the target space. Each vector in the basis contributes a brand-new dimension, ensuring no redundancy, while collectively they can reach every conceivable point in the space.

Conditions: The set of vectors must be linearly independent.; The set of vectors must span the target space.

The video reinterprets a coordinate pair like $(3, -2)$ not merely as a static location on a grid, but as dynamic scaling factors. The first number scales the horizontal unit basis vector $\hat{i}$, and the second number scales the vertical unit basis vector $\hat{j}$.

Conditions: Working in a standard 2D Cartesian coordinate system.; Understanding $\hat{i}$ and $\hat{j}$ as the standard unit basis vectors.