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Numerical Linear Algebra with Applications by William Ford

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Chapter 3

Subspaces

Abstract

This chapter introduces necessary vector and matrix properties. It is somewhat abstract, but should be studied thoroughly. The chapter defines the concept of a subspace of a vector space and uses the null space of matrix and the span of a set of vectors as examples. In addition, the row and column spaces of a matrix are presented. A discussion of linear independence and a basis gives rise to the notion of the dimension of a subspace. It is stated that the rank of a matrix is the dimension of its row space, and the nullity of a matrix is the dimension of its null space. The chapter notes that the dimension of the column space equals the dimension of the row space, and that rank(A) + nullity(A) = n for an n ×n

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