Linear Independence, Span and Basis
Span is everything you can reach by combining vectors; independence says none of them is redundant. A basis is a set that is both, so every element has exactly one expression.
What a learner can do afterwards
- Decide independence by solving a homogeneous system
- Describe the span of a set of vectors geometrically
- Show that coordinates with respect to a basis are unique
1 · Read
Vectors are independent when only the trivial combination gives zero: if scaled copies add to zero, every scale must be zero. Test this by solving the homogeneous system. A shortcut for pairs: if one vector is a multiple of the other, the pair is dependent. Merely avoiding the zero vector is not enough.
The span of a set is everything you can build by combining its members. One nonzero vector spans a line through the origin in its direction. Two vectors that are not multiples span a whole plane. Since two already span the plane, any third vector in it depends on them, and three is the minimum that can span all of three dimensional space.
A basis is an independent set that spans the whole space. The pair (1, 0) and (0, 1) is a basis of the plane. The vector (5, 7) is 5 times (1, 0) plus 7 times (0, 1), and independence makes this the only way. Three vectors in the plane can never form a basis: one of them always depends on the rest.
To show coordinates are unique, suppose two writings agree and subtract them. The difference is a combination giving zero, so independence forces every coefficient to match. That is why each vector has exactly one expression in a basis. Remember the chain: solve the system, read the geometry, then trust the basis.
Independence means no redundancy, span means full reach, and a basis gives both, so coordinates are unique.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.