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10/23/04 2:34 AM
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jgibson
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Edited: 23-Oct-04
Member Since: 04/30/2001
Posts: 4363
 
Alright, I'm not looking for the answer to this...but a hint would help. Consider the vector subspace W of R^n as P(n)=1 + x + x^2 + ... x^n. Derive the hierarchy of vector subspaces up to the vector space V=R^n that gives you P(n).
10/30/04 9:55 PM
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quamrh
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Edited: 30-Oct-04
Member Since: 01/01/2001
Posts: 194
If I need a 40% antifreeze mixture and have 20L of solution at 20% how much do I need to drain and add 100% to in order to get the 20L @ 40%?
11/1/04 10:59 PM
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jgibson
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Edited: 01-Nov-04
Member Since: 04/30/2001
Posts: 4379
Cajones, I think I've got it down. I'm gonna give it my prof and see what he thinks, then I'll post.
11/8/04 1:22 AM
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jgibson
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Edited: 08-Nov-04
Member Since: 04/30/2001
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Recall from calculus that if f(x) and g(x) are continuous functions on the interval (-inf,inf) and k is a constant, then f + g and kf are also continuous. Thus, the continuous functions on the interval (-inf,inf) form a subspace F(-inf,inf), since they are closed under addition and scalar multiplication. We will denote this subspace by C(-inf,inf). Similarly, if f and g have continuous first derivatives on (-inf,inf), then so do f + g and kf. Thus, then functions with continuous first derivatives on (-inf,inf) form a subspace of F(-inf,inf). We denote this subspace C^1(-inf,inf), where the superscript 1 is used to emphasize first derivative. However, it is a theorem of calculus that every differentiable function is continuous, so C^1 is actually a subspace of C(-inf,inf). Take this a a step further, and for every positive integer m, the functions with continuous mth derivatives on (-inf,inf) form a subspace of C^1(-inf,inf) as do the functions that have continuous derivates of all orders. We denote the subspace of functions with continuous mth derivatives on (-inf, inf) by C^m(-inf,inf), and we denote the the subspace of funcions that have continuous derivatives of all orders on (-inf,inf) by C^inf(-inf,inf). Finally, it is a theorem of calculus that polynomials have continuous derivatives of all orders, so P_n is a subspace of C^inf(-inf,inf). Whew...there it is.
11/8/04 11:42 AM
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jgibson
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Edited: 08-Nov-04
Member Since: 04/30/2001
Posts: 4391
LOL
11/9/04 6:33 PM
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asdf
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Edited: 09-Nov-04
Member Since: 01/01/2001
Posts: 6526
great caesar's ghost!

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