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Find a basis for this vector space. c. Show that differentiating a polynomial is a linear transformation. d. Given the basis chosen in part (b), write down the matrix representative of the derivative. 3. Let A and B be two n x n invertible matrices. Prove that 4. Let A = (; ~) Find a matrix C so that C-1 AC is a diagonal matrix. 5. Denote the vector space of all functions f:R-+R which are infinitely differentiable by COO(R). This space is called the space of smooth functions. a. Show that COO(R) is infinite dimensional.

Continuous or integrable functions) does not guarantee the reasonableness of the limit, as we will see in the next example. Here we show that the pointwise limit of continuous functions need not be continuous. For each positive integer n, set for all x on [0,1]. CHAPTER 2. 36 Set f(x) = {~: € AND 8 REAL ANALYSIS x=1 O:S;x<1 • Clearly f(x) is not continuous at the endpoint x = 1 while all of the functions fn(x) = x n are continuous on the entire interval. But we will see that the sequence Un(x)} does indeed converge pointwise to f(x).

For example, if the above definition of a limit is to make sense, it must yield that lim x 2 = 4. x-t2 We will check this now. It must be emphasized that we would be foolish to show that x 2 approaches 4 as x approaches 2 by actually using the definition. We are again doing the common trick of using an example whose answer we already know to check the reasonableness of a new definition. Thus for any E > 0, we must find a 0 > 0 so that if 0 < Ix - 21 < 0, we will have Set o= min ( i, 1). As often happens, the initial work in finding the correct expression for 0 is hidden.

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All the Mathematics You Missed [college level,bw] by Garrity


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