AcademyVector Spaces
Academy
Spanning Sets
Level 1 - Math I (Physics) topic page in Vector Spaces.
Principle
The span of a list of vectors is the set of all linear combinations of those vectors. If a list spans a vector space, every vector in the space can be built from that list.
In physics, a spanning set describes all states, motions, or component combinations a linear model can produce. If the available directions do not span the target space, some physical configurations cannot be represented.
Notation
\(\operatorname{span}\)
the set of all linear combinations of a list of vectors
\(\mathbf v_1,\ldots,\mathbf v_k\)
the vectors being used to form combinations
\(c_i\)
a scalar coefficient multiplying \mathbf v_i
\(\mathbf x\)
a target vector to test against the span
\(c_1\mathbf v_1+\cdots+c_k\mathbf v_k\)
a general linear combination
A span always contains the zero vector because every coefficient can be chosen as zero.
Method
- Write the target vector as \(\mathbf x=c_1\mathbf v_1+\cdots+c_k\mathbf v_k\).
- Convert the vector equation into simultaneous equations for the coefficients.
- Solve the system.
- If a solution exists, \(\mathbf x\) lies in the span.
- To prove the list spans a whole space, show that every possible target vector in that space can be written this way.
Rules
Span definition
\[\operatorname{span}\{\mathbf v_1,\ldots,\mathbf v_k\}=\{c_1\mathbf v_1+\cdots+c_k\mathbf v_k:c_i\in\mathbb R\}\]
Standard span
\[(x,y)=x(1,0)+y(0,1)\]
One-vector span
\[\operatorname{span}\{(1,2)\}=\{t(1,2):t\in\mathbb R\}\]
- A span is always a subspace.
- Adding a vector that is already in the span does not change the span.
- The standard coordinate vectors span \(\mathbb R^n\).
- One non-zero vector spans a line through the origin, not the whole plane or three-dimensional space.
Examples
Question
Why do
\[(1,0)\]
and \[(0,1)\]
span \[\mathbb R^2\]
?Answer
Take any vector
\[(x,y)\in\mathbb R^2\]
It can be written as \[x(1,0)+y(0,1)=(x,0)+(0,y)=(x,y)\]
Since every vector \[(x,y)\]
has this form, the two standard vectors span \[\mathbb R^2\]
Checks
- To prove a list spans a whole space, test a general target vector, not one convenient vector.
- Keep coefficient variables separate from vector components.
- Remember that spans pass through the origin.
- A spanning set may contain redundant vectors; spanning alone does not imply independence.