 
Summary: 18.024ESG Notes 1
Pramod N. Achar
Spring 2000
The goal of this set of notes is to make sense of the idea of "dimension" of linear subspaces of Rn
. An
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
