Define two functions T: ℝ^2 ⟶ ℝ^3 and S: ℝ^2 ⟶ ℝ^2 by T [X, Y] = [2x + y, 0], S[x,y] = - [x +y, xy] Determine whether T and S, and the composite S ∘ T are linear transformations.
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Define the transformation S: RR by S(x, y) = (1 - Ty, x + y). (a) (6 points) Determine whether S is a linear transformation. Support your answer with calculations and explanation. (b) (2 points) Find the image of ū = (5,2)
Determine whether the following transformations are linear. A) T(x, y) = (3x, y, y ? x) of R2 ? R3 B) T(x, y) = (x + y, 2y + 5) of R2 ? R2
1. Let B-(0, 1). Define x + y max(x, y) and x . y-min(x, y), and let the complement of x of be 1-x (ordinary subtraction). Show whether or not B forms a Boolean algebra under these operations. 2. Let S-(0,1 R, and T = { y : 2 < y < 12). Find a one to one correspondence (the actual function) between S and T showing they have the same cardinality. (hint: look at straight lines in the xy-plane)...
Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute the values (1), y(2) Consider the system: z'(t) + tr(t) + (t-1 )y(t) = 0, s(t) + (t-1)x(t) + ty(t) = 0, x(0)--4 y(0) = 2 Determine the solution functions, ()y) using ONLY the Fundamental Matrix method. Compute the values (1), y(2)
Define T : R3 → R2 by T(x,y,z) = (2x +4y +3z,6x) Show that T is linear.
The functions æ(t), y(t) satisfy the system of equations x (t) = -3 x (t) – y(t) ft y(t) = 5 x (t) – y(t) and the initial conditions x(0) = 1 and y(0) = -1. Suppose that the Laplace transforms of x(t), y(t) are respectively X(s), Y(s). By forming algebraic equations in X(s), Y(s)., find and the enter the function X(s), Y(s) below in maple syntax. X(s) = Y(s) =
Problem 4. Given the input/output system represented by t-1 y(t) = 2 ( x(y - 3) dy where x(t) is the input and y(t) is the output, a) Determine whether the system is linear or non-linear. b) Determine the impulse response h(t, to) of the system by setting x(t)= 8(t–to). c) Determine whether the system is time invariant or time variant. d) Determine whether the system is causal or non-causal.
Linear Algebra! Practice exam #1 question 1 Thanks for sloving! 1- Transformations (3 points each) a) Given a linear transformation T :N" N" T(x,y)-(x-y,x+y) and B= {< l, 0>.< 1,1 >} , B = {< l, l>,< 0, l>} V,-< 2, l> Find V,T,and TVg) b) Given a linear transformation T:n'->n2 T(x,y,2)-(x-z,x +2y)and V =< 2,-I, I> B= {<l, 0, 1>.< 1, 1, 0 >, < 0, l, 0 >}, B' = {<l, l >, < 0, 1 >} Find...
Given two functions, M(x, y) and N(x,y), suppose that ON/ that an/az-amay is M-N a function of x +y. That is, let f(t) be a function such that ON _ OM dc du f(x+y) = M-N Assume that you can solve the differential equation Mdx + Ndy = 0 by multiplying by an integrating factor u that makes it exact and that it can also be written as a function of x + y, u = g(x + y) for...
1. Use combinations of STEP FUNCTIONS to describe each continuous-time signal shown below. f(t) 0 2 4 6 0 1 2 3 0 1 2 3 4 2. Sketch the following signals: (a) x (t)=1 [u(t+2)-u(t-1)] (c) X(t)=\fety (b) X(t)=t.e (d) x (t) = u(t) u(t-1).ult-2).u(t-3) 3. Determine whether the systems below are linear and time invariant. Justify your answer! (a) y(t) = x(31) (b) y(t)= 2x(1-t) y(t)=cos(x(t)] 4. Simplify the expressions: (a) y(t)=1.8(t+2)+(t +1) 8(1-1)+(t+3). 8(t) (b) y(t) =...