Vector Spaces

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In Chapter ?? we discussed how to solve systems of $m$ linear equations in $n$ unknowns. We found that solutions of these equations are vectors $(x_1,\ldots ,x_n)\in \R ^n$ . In Chapter ?? we discussed how the notation of matrices and matrix multiplication drastically simplifies the presentation of linear systems and how matrix multiplication leads to linear mappings. We also discussed briefly how linear mappings lead to methods for solving linear systems — superposition, eigenvectors, inverses. In Chapter ?? we discussed how to solve systems of $n$ linear differential equations in $n$ unknown functions. These chapters have provided an introduction to many of the ideas of linear algebra and now we begin the task of formalizing these ideas.

Sets having the two operations of vector addition and scalar multiplication are called vector spaces. This concept is introduced in Section ?? along with the two primary examples — the set $\R ^n$ in which solutions to systems of linear equations sit and the set $\CCone$ of differentiable functions in which solutions to systems of ordinary differential equations sit. Solutions to systems of homogeneous linear equations form subspaces of $\R ^n$ and solutions of systems of linear differential equations form subspaces of $\CCone$ . These issues are discussed in Sections ?? and ??.

When we solve a homogeneous system of equations, we write every solution as a superposition of a finite number of specific solutions. Abstracting this process is one of the main points of this chapter. Specifically, we show that every vector in many commonly occurring vector spaces (in particular, the subspaces of solutions) can be written as a linear combination (superposition) of a few solutions. The minimum number of solutions needed is called the dimension of that vector space. Sets of vectors that generate all solutions by superposition and that consist of that minimum number of vectors are called bases. These ideas are discussed in detail in Sections ??–??. The proof of the main theorem (Theorem ??), which gives a computable method for determining when a set is a basis, is given in Section ??. This proof may be omitted on a first reading, but the statement of the theorem is most important and must be understood.