Spanning Sets
A Spanning Set is the smallest set of vectors where any vector in a certain space can be represented as a Linear Combination of the spanning set vectors.
The Spanning Set can span a line (1-dimension), a plane (2-dimension) or a space (3-dimensions).
Look how the Standard Basis unit vectors can be used to create Spanning Set:
| Spanning Set | Line | Spanning Set | Plane | Spanning Set | Space |
| \(\hat{\imath}\) | \(x\)-dir | \(\hat{\imath}\), \(\hat{\jmath}\) | \(xy\)-plane | \(\hat{\imath}\), \(\hat{\jmath}\), \(\hat{k}\) | 3D space |
| \(\hat{\jmath}\) | \(y\)-dir | \(\hat{\imath}\), \(\hat{k}\) | \(xz\)-plane | ||
| \(\hat{k}\) | \(z\)-dir | \(\hat{\jmath}\), \(\hat{k}\) | \(yz\)-plane |
Given a spanning set of just \(\hat{\imath}\), the only vectors we can make using linear combnations are vectors along the x-axis. Therefore, it spans a line.
\(2\hat{\imath}\),\(-1.5\hat{\imath}\), etc.
Given a spanning set of \(\hat{\jmath}\), \(\hat{k}\), the only vectors we can make using linear combinations are vectors in the yz-plane. Therefore, they span a plane.
\(2\hat{\jmath} + \hat{k}\), \(-2\hat{\jmath} + \hat{k}\), etc.
Given a spanning set of \(\hat{\imath}\), \(\hat{\jmath}\), \(\hat{k}\), we can make any vector in 3D space using linear combinations. Therefore, they space a space.
\(\hat{\imath} + \hat{\jmath} - 0.5\hat{k} \), etc.
Let's see a few more examples:
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| The lines are Coplanar. They both lie on a plane. | The lines are Coplanar. They all lie on a plane. | The lines are non-Coplanar. They all cannot lie on the same plane. |
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| Spans a plane. | Span a plane. | Span a space. |