Why is the span of a collection considered the smallest containing subspace?
Conditions
- The collection of vectors is fixed.
- We are comparing subspaces that contain this collection.
Reasoning, step by step
- Define the span as the set of all linear combinations of the collection.
- Note that any subspace containing the collection must be closed under addition and scalar multiplication.
- Conclude that such a subspace must therefore contain all linear combinations of the collection.
- Identify the span as the minimal set satisfying these closure properties.
Example
The span of a collection consists of all its linear combinations. It is a subspace containing the collection, and every other subspace containing those vectors must also contain their span.
Common misconceptions
- Thinking that there could be a smaller subspace that contains the vectors but not all their linear combinations.
- Confusing the span with the set of vectors itself.
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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 (), 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
The span of a collection consists of all its linear combinations. It is a subspace containing the collection, and every other subspace containing those vectors must also contain their span.
Conditions: The collection consists of one or more vectors.; Linear combinations involve scalar multiplication and vector addition.
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.
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