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In 3D space, the span of two non-parallel vectors forms a tilted, infinitely extending flat sheet cutting through the origin. While each individual vector spans a line, their linear combination $a\vec{v} + b\vec{w}$ traces out this entire 2D plane embedded within the 3D volume.

Conditions: Working in three-dimensional Euclidean space; Two vectors $\vec{v}$ and $\vec{w}$ are non-parallel

In 3D space, the span of two non-parallel vectors forms a tilted, infinitely extending flat sheet cutting through the origin. While each individual vector spans a line, their linear combination $a\vec{v} + b\vec{w}$ traces out this entire 2D plane embedded within the 3D volume.

Conditions: Working in three-dimensional Euclidean space; Two vectors $\vec{v}$ and $\vec{w}$ are non-parallel

Nonzero collinear vectors span a line, because their linear combinations can only reach points along that single shared line. An all-zero set spans only the origin, as any linear combination of zero vectors results in the zero vector.

Conditions: For the first case, the vectors are nonzero and collinear.; For the second case, all vectors in the set are the zero vector.

Nonzero collinear vectors span a line, because their linear combinations can only reach points along that single shared line. An all-zero set spans only the origin, as any linear combination of zero vectors results in the zero vector.

Conditions: For the first case, the vectors are nonzero and collinear.; For the second case, all vectors in the set are the zero vector.