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
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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.
Solve a(1, 0, 1) + b(0, 1, 1) + c(1, 1, 2) = (0, 0, 0). The entries give a + c = 0, b + c = 0 and a + b + 2c = 0. The first two make a = b = -c, and then the third reads 0 = 0, so c is free. With c = 1, a = b = -1: a nontrivial combination, so the three vectors are dependent. Since (1, 1, 2) = (1, 0, 1) + (0, 1, 1), they span only a plane through the origin.
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 combinations of the basis give the same vector, 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.
Independence means no redundancy, span means full reach, and a basis gives both, so coordinates are unique.
Reading notes describe open textbooks listed on OER Commons: A First Course in Linear Algebra (LibreTexts), Understanding Linear Algebra, and Vector Spaces First (University of Ottawa). Videos by Khan Academy. Practice questions are original.
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