 # Linear Algebra Question

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).

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%?

Cajones,

I think I've got it down. I'm gonna give it my prof and see what he thinks, then I'll post.

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.

LOL

great caesar's ghost!