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lec2 - MAT1341A LECTURE NOTES FOR FALL 2008 MONICA...

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MAT1341A: LECTURE NOTES FOR FALL 2008 MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP) 4. Vector Spaces Last week, we established that it wasn’t too hard to generalize our notion of “vectors” to include elements of R n . We still call them “vectors” even though we can no longer quite imagine them as arrows in space. As we worked through vector geometry last week, we also saw a number of problems coming up, principally among them: what are the higher-dimensional analogues of lines and planes? how can you describe them? We will tackle this problem next, but it turns out that the best way to do this is to step back and see just how far we can generalize this notion of “vector spaces” — was R n the only set that “behaves” substantially like R 2 and R 3 , or are there many more mathematical objects out there that, if we look at them in the right way, are just like vectors, too? The main part of the next three lectures comes from [N, Ch 5.1], but some of the results also appear in [N, Ch 4.1], and we use some examples from [N, Ch 1.1]. 4.1. A first example. So far: we agree that the elements of R , R 2 and R 3 are “geometric vectors”. We also agree that we can call elements of R n , for n 4, “vectors”. What we are looking for next: non-geometric, non- R n vectors. Example: Spaces of Equations Consider three equations, which we name E 1 , E 2 and E 3 : E 1 : x - y - z = - 1 E 2 : 2 x - y + z = 1 E 3 : - x + 2 y + 4 z = 4 We can create new equations from these ones; for example, E 4 = E 2 - 2 E 1 : y + 3 z = 3 or E 1 + E 3 : y + 3 z = 3 Date : September 16, 2008. 1
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2 MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP) So we can even say E 2 - 2 E 1 = E 1 + E 3 and it is legitimate to rewrite this as: 3 E 1 - E 2 + E 3 = 0 where “0” stands for the equation “0 = 0”. Well, what are we really saying here? We can add two equations to get another equation. We can multiply and equation by a scalar to get another equation. There’s a zero equation given by 0 = 0. Let’s call it E 0 . Equations have negatives : - E 1 : - x + y + z = 1 Why is this the negative? Because now ( E 1 ) + ( - E 1 ) = E 0 . The usual rules of arithmetic hold: E 1 + E 2 = E 2 + E 1 E 1 + ( E 2 + E 3 ) = ( E 1 + E 2 ) + E 3 k ( E 1 + E 2 ) = kE 1 + kE 2 ( k + l ) E 1 = kE 1 + lE 1 ( kl ) E 1 = k ( lE 1 ) 1 E 1 = E 1 In other words: these are exactly the properties of R n that we identified last week! So even though we have no reasonable way (yet) of writing “equations” as n -tuples of numbers, algebraically we recognize that they act just like vectors do. Major idea #1 (today’s lecture): there are lots of different vector spaces besides R n . We can specialize this a bit more (which also helps us start to see the value in this approach): Consider the space E of all equations “obtainable from” (also say: “generated by” or “spanned by”) E 1 , E 2 , and E 3 : E = { k 1 E 1 + k 2 E 2 + k 3 E 3 | k i R } .
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