for each matrix A below, describe the invariant subspaces for
the induced linear operator T on...
for each matrix A below, describe the invariant subspaces for
the induced linear operator T on F^2 that maps each v set of F^2 to
T(v)=Av. (a) [4,-1;2,1], (b) [0,1;-1,0], (c) [2,3;0,2], (d)
[1,0;0,0]
1. For each matrix A below compute the characteristic polynomial
χA(t) and do a direct matrix computation to verify that χA(A) =
0.
(4 3
-1 1) (2 1 -1 0 3 0 0 -1 2) (3*3 matrix)
2. For each 3*3 matrix and each eigenvalue below
construct a basis for the eigenspace Eλ.
A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3
A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2...
Let T be a linear operator on a finite-dimensional complex
vector space V . Prove that T is diagonalizable if and only if for
every λ ∈ C, we have N(T − λIV ) = N((T − λIV
)2).
Consider the linear time invariant system described by the
transfer function G(s) given below. Find the steady-state response
of this system for two cases: G(s) = X(s)/F(s) =
(s+2)/(3(s^2)+6s+24) when the input is f(t) = 5sin(2t) and f(t) =
5sin(2t) + 3sin(2sqrt(3)t)
Question 2. True or false
2. If [T]β is the matrix representing the linear map T in the
basis β, then the jth column of [T]β contain the coordinates of the
T(βj) in the basis β, and same for the rows.
Using MATLAB, determine whether the system below are a)
linear/non-linear b) time-invariant/timevariant, c)
causal/noncausal, d) has memory/memoryless:
y(t) = x(t) + x(t -1)
Provide MATLAB code and graphs to show your work for the
linearity and time-invariance testing.
For each situation below determine the direction of the induced
current in the loop (if there is one).
(a) A square loop is moving at a constant velocity to the right
through a uniform magnetic fieldthat is directed into the page and
which extends out of the picture to the left and right. In which
direction is the induced current in the loop?
[ ] clockwise [ ] counter-clockwise [ ] there is no induced
current Justify your answer (with...
Classify the following system with output current given (by the
equation below) as linear/non linear
i0(t)= 5 i1(t) + 8 i2 (t-10) + 0.25 i3 (t+2)
Is this linear or non linear
T::R2->R2, T(x1,x2) =(x-2y,2y-x). a) verify that this
function is linear transformation. b)find the standard matrix for
this linear transformation. Determine the ker(T) and the range(T).
D) is this linear combo one to one? how about onto? what else could
we possibly call it?
For each of the situations below, write three different
equations for x(t) that all describe the same
motion. One of your three equations must use sine, one of your
three equations must use cosine, and at least one must use a
non-zero phase shift (ϕ), whose value you should specify.
A mass-spring system of angular frequency ω is stretched to
x=+A and released. The time is started so that
t=0 is when the mass passes through the equilibrium
position at...