What constitutes the span of a collection of vectors?

A third vector lying perfectly flat on the plane created by the first two provides no new directional freedom. Mathematically, it is redundant because it can be written as a linear combination of the other two vectors.

Conditions: Three vectors exist in 3D space; The first two vectors are linearly independent (non-parallel); The third vector lies within the plane spanned by the first two

Adding a third vector that pokes out of the plane at an angle 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 expands from a plane to encompass every conceivable point in the 3D volume, effectively filling all of 3D space.

Conditions: Starting with two linearly independent vectors spanning a plane; Introducing a third vector that is NOT in that plane

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)

The set is considered a subspace because vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin.

Conditions: The vectors are of the form (x,0,-x).; The operation is vector addition or scalar multiplication.

A linear combination of two arbitrary non-collinear vectors $\vec{v}$ and $\vec{w}$ is formed by multiplying each vector by a scalar ($a$ and $b$) and adding them together ($a\vec{v} + b\vec{w}$). If both scalars are allowed to vary continuously across all real numbers, the tips of these combined vectors sweep out every possible point on the 2D plane.

Conditions: The two vectors $\vec{v}$ and $\vec{w}$ are non-collinear (point in different directions and do not lie on the same line).; The scalars $a$ and $b$ vary continuously across all real numbers.