CDS 212 - Solutions to Homework # 1 Instructor
نویسنده
چکیده
Problem 1 (a) Recall that the null space and the range of a matrix A ∈ C m×n are defined as: N (A) = {x ∈ C n |Ax = 0} R(A) = {y ∈ C m |y = Ax, for some x ∈ C n } Note that N (A) is a subspace of C n since it contains 0 ∈ C n (A0 = 0) and is closed under addition and scalar multiplication: x 1 , x 2 ∈ N (A) ⇒ Ax 1 = 0 and Ax 2 = 0 ⇒ Ax 1 + Ax 2 = 0 ⇒ A(x 1 + x 2) = 0 ⇒ x 1 + x 2 ∈ N (A) x ∈ N (A), α ∈ C ⇒ A(αx) = αAx = 0 ⇒ αx ∈ N (A) Similarly, note that R(A) is a subspace of C m since it contains 0 ∈ C m (A0 = 0) and is closed under addition and scalar multiplication:
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CDS 212 - Solutions to Homework # 2
(a) [Problem 2.2 in DFT] For u 1 : For u 2 : For u 3 : Note that u 1 = u 2 2 = ∞ 0 1 · dt = ∞, while u ∞ = 1 and P ow(u) = lim T →∞ 1 2T T 0 1 · dt = 1 2. T −T u 2 (t)dt = T 0 u 2 (t)dt = T 0 |u(t)|dt = k−1 i=0 2 2i for 2 2k−1 ≤ T ≤ 2 2k , k ≥ 1. Hence, u 1 = u 2 = ∞ while u ∞ = 1. Finally: P ow(u) = lim T →∞ 1 2T T −T u 2 (t)dt = lim k→∞ 1 2 2k k−1 i=0 2 2i = 1 4 when T = 2 2k−1 , k ≥ 1 ...
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