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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.

All vector spaces have to obey the eight reasonable rules. A real vector space is a set of “vectors” together with rules for vector addition and multiplication by real numbers. The addition and the multiplication must produce vectors that are in the space. And the eight conditions must be satisfi ed (which is usually no problem). You need ...

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