What is the span of three vectors in 3D space when the third vector pokes out of the plane?

In linear algebra, a coordinate pair like $(3, -2)$ is viewed not just as a static location, but as scaling factors. The first number scales the horizontal unit vector $\hat{i}$, and the second scales the vertical unit vector $\hat{j}$.

Conditions: Working within a standard 2D Cartesian coordinate system; Using the standard unit basis vectors $\hat{i}$ and $\hat{j}$

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)

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.