Summary: 18.024ESG Notes 1
Pramod N. Achar
The goal of this set of notes is to make sense of the idea of "dimension" of linear subspaces of Rn
intuitive idea of dimension might say that it is the number of parameters needed to specify a point, or the
number of degrees of freedom, in some sense. To make this idea formal, we have introduced the notions of
spanning set and linear independence. Roughly, the idea of a spanning set corresponds to having enough
parameters to describe points in the space, and the idea of linear independence corresponds to nonredundancy
among those parameters.
Let W Rn
be a linear subspace. A given set of vectors v1, . . . , vk W might span W without being
linearly independent (i.e., they might be redundant); or, they might be linearly independent but not span
W. (Of course, they might also be neither linearly independent nor a spanning set.) A spanning set that
is not linearly independent is too large, and a linearly independent set that does not span is too small. A
basis for a linear subspace is a linearly independent spanning set. The size of a basis should be "just right"
to describe points in the subspace, in the following sense:
· Every vector in the subspace can be written as a linear combination of the basis vectors (because the
basis is a spanning set).
· There is only one way of writing a given vector as a linear combination of basis vectors (because they