s09_mthsc208_hw18

# s09_mthsc208_hw18 - MTHSC 208(Diﬀerential Equations Dr...

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Unformatted text preview: MTHSC 208 (Diﬀerential Equations) Dr. Matthew Macauley HW 18 Due Monday March 30th, 2009 (1) For each of the following sets, determine if it is a vector space. If it is, give a basis. If it isn’t, explain why not. (a) The set of points in R3 with x = 0. (b) The set of points in R2 with x = y . (c) The set of points in R3 with x = y . (d) The set of points in R3 with z ≥ 0. (e) The set of unit vectors in R2 . (f) The set of polynomials of degree n. (g) The set of polynomials of degree at most n. (h) The set of polynomials of degree at most n, with only even-powers of x. 2 ∞ (i) The set of generalized power series of the form n=0 an xn− 3 . (j) The set of 2π -periodic functions. (2) Complete the following sentences: (a) Two non-zero vectors v1 and v2 are a basis for R2 if and only if. . . (b) Three non-zero vectors v1 , v2 , and v3 are a basis for R3 if and only if. . . (3) Let X be a vector space over C (i.e., the contants are complex numbers, instead of just real numbers). If {v1 , v2 } is a basis of X , then by deﬁnition, every vector v can be written uniquely as v = C1 v1 + C2 v2 . (a) Is the set {v1 + v2 , 3v1 − 2v2 } also a basis of X ? 1 1 1 (b) Is the set { 2 v1 + 1 v2 , 2i v1 − 2i v2 } a basis of X ? 2 (c) Consider the ODE y + 4y = 0. If we assume that y (t) = ert , then we get that r = ±2i. Therefore, the general solution is y (t) = C1 e2it + C2 e−2it , i.e., {e2it , e−2it } is a basis for the solution space. Use (b), and Euler’s equation (eiθ = cos θ + i sin θ) to ﬁnd a basis for the solution space involving sines and cosines, and write the general solution using sines and cosines. ...
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