Consider T:R4 → R3 with 1 [T) 3 2 -4-1 0 0 5 7 8 10 4 3 -20 -6 1 0 0 5 om Find a basis for image(T). What is dim(image(T))? Convert image(T) to relation form.
consider T(x)= A (x) [1 2 0 3 6 1 2 4 1 1 2 3 -1 2 9 2 1 5 10 11 0 (a) Find a basis for the nullspace (kernel) of T. (b) Find a basis for the range of T. (c) What are the values of the rank and nullity?
2. Consider a periodic signal shown below (20 points) i(t) -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 (a) Find the fundamental period of this signal. (b) Consider following two signals, xi(t) and x2(t), obtained from the above signal, find their corresponding Fourier transforms: xi(t) -1 0 1 *2 (1) Its of, 0 1 2 3 4 5
4. (20 points) Consider the periodic signal r(t) shown in the Figure below: x(t) 3 2 N VAA 0 1 2 3 4 5 6 A . Determine the fundamental period T and the fundamental frequency wo. B. Compute the Fourier Series coefficients and simplify the expression to its simplest form.
4 1 1. The matrix A and it reduced echelon form B are given below. 1-2 95 4 10 3 0 0 1 -1 6 5 3 0 1 -3 0 -7 -2 0 -6 1 -2 0 0 0 1 -2 9 1-9 0 0 0 0 0 (a) Find p, q, r s t Nul A, Col A, Row A is a subspace of RP, R9, R', respectively Answer. p = 9= (b) Find a basis for Nul...
Problem 5 Consider the linear system [1 2 0 2 -4 7x(t) 1 -4 6 y(t) [1 -2 2] (t). (4) a(t = (a) Is the system (4) observable? (b) Give a basis for the unobservable subspace of the system (4). In the remainder of this problem, consider the linear system а — 3 8— 2а 0 1 2a u(t) (t) (5) x(t) = with a a real parameter. (c) Determine all values of a for which the system (5)...
1 2 -3 1 -6 -2 5 2. 4. (10 points) Let A = (a) (5 points) Find a basis for col(A) and calculate rank(A). (b) (5 points) Find a basis for null(A) and calculate nullity(A).
6. Consider the recurrence relation T(n) = 2T(n-1) + 5 for integers n 1 and T(O) = 0. Find a closed-form solution Using induction, prove your solution correct for all integers n 20.
Problem 5: (20=10+5+5 points) Consider a causal system with the following transfer function (the z-transform) 2 +42-2 H2) = 1 - 52-1 +6 1-52-1 +62- 2 2 < 121 <3. (a) Find the inverse z-transform of H(2). (b) Find the difference equation with input-output relation for this system. (c) Draw the diagram of realization (in form II) for this system.
1. Consider the matrix 12 3 4 A 2 3 4 5 3 4 5 6 As a linear transformation, A maps R' to R3. Find a basis for Null(A), the null space of A, and find a basis for Col(A), the column space of A. Describe these spaces geometrically. 2. For A in problem 1, what is Rank(A)?