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lec4_456 - NOTES ON SECTION 4.4 4.6 MA265 Some key words...

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NOTES ON SECTION 4 . 4 - 4 . 6 MA265, SECTIONS 41, 52 Some key words basis dimension linearly dependent linearly independent span spanning set 1. Overview We want two things in this set of section: A measure for how large a vector space is (ultimately provided by dimension). A system to parameterize elements in a vector space by coordinates (provided by a basis). They are related: bases are used to calculate dimension. But you should also understand that bases do much more than that: they provide a coordinate system that allows one to parameterize the vector space by real numbers, thus allowing us to compare the vector space to R n . We don’t pursue this as far as possible, but I strongly encourage you to at least read section 4 . 8 to get some of the details of what I mean by this in a more precise fashion. As we will see, dimension is a count of the number of parameters needed to describe a vector space (alternative, the number of independent directions it has). There are two questions to ask (1) Given a collection of parameters (directions) describing elements in the vector space, we ask: “Do I have enough parameters (directions)?” (2) We also ask: “Do I have too many parameters (redundant or dependent directions)?” The first question is clarified by the notion of “span”, the second by the notion of ”linear in/depenendence”. 2. Span Definition 1. We defined span already in 4 . 3 . Definition 2. A set of vectors spans V if span ( S ) = V . In this case S is called a spanning set for V . Example 3. Take V = R 3 . The set of vectors S = 1 0 0 , 0 1 0 , 0 0 1 spans R 3 . You can make the obvious generalization to get spanning sets of R n in general. Example 4. Take V = M 2 × 2 , the set of 2 × 2 matrices with real coefficients (itself a vector space). It has the following spanning set S = 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 Example 5. If S is a spanning set for a vector space V , so is any set S 0 which contains S : the reason is that if you can build any vector out of the vectors in S , and everything in S is in S 0 , then surely you can build any vector out of the vectors in S 0 (just forget about the extra vectors you added to get from S to S 0 . For instance, in Example 3 the set S 0 S 0 = 1 0 0 , 0 1 0 , 0 0 1 , 1 1 0 , 0 1 - 1 , 1 2 1 also spans. 1
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In this way, span allows us to determine if we have enough vectors: enough means that we can build any vector out of the vectors in S using linear combinations. But it doesn’t tell us if we have too many (which is why we’ll need linear dependence later). It is useful to have a criteria in R n to determine if a set of vectors spans R n : here is the answer Theorem 6. If S is a set of vectors, and A is the matrix obtained by putting the vectors in S as the columns of a matrix, then S spans R n if and only if the RREF of A has no row of zeros.
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