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Vector Spaces and Subspaces - MIT Mathematics

Chapter 5 Vector Spaces and The Column Space of a MatrixTo a newcomer, matrix calculations involve a lot of you, they involve columns ofAvandABare linear combinations ofnvectors the columns ofA. Thischapter moves from numbers and vectors to a third level of understanding (the highestlevel). Instead of individual columns, we look at Spaces of vectors. Without seeingvector spacesand theirsubspaces, you haven t understood everything this chapter goes a little deeper, it may seem a little harder. That is natural. Weare looking inside the calculations, to find the Mathematics . The author s job is to make itclear. Section will present the Fundamental Theorem of Linear Algebra. We begin with the most important Vector Spaces . They are denoted byR1,R2,R3,R4,: : :. Each spaceRnconsists of a whole collection of all columnvectors with five components.

Vector Spaces and Subspaces 5.1 The Column Space of a Matrix To a newcomer, matrix calculations involve a lot of numbers. To you, they involve vectors. The columns of Av and AB are linear combinations of n vectors—the columns of A. This chapter moves from numbers and vectors to a third level of understanding (the highest

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