18.024ESG Notes 1 Pramod N. Achar 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 Collections: Mathematics