Question

For the linear operator T on V , find an ordered basis for the T -cyclic subspace generated by the vector v.

(a) V =R3,T(a,b,c)=(2a+b,b−3a+c,4c)wherev=(1,0,0).

(b) V = P3(R), T(f(x)) = 3f′′(x), and v = x2.

Answer #1

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For each of the following linear operators T on vector space V,
compute the determinant T and the characteristic polynomial of
T.
(a). V = R2 , T(a, b) = (2a - b, 5a + 3b)
(b). V = R3 , T(a, b, c) = (a - 3b + 2c, -2a + b + c,
4a - c)
(c). V = P3(R) , T(a, b, c) = T(a + bx +
cx2 + dx3) = (a - c) + (-a...

Find eigenvectors associated with the eigenvalues of the following
linear operator: (please show all steps)
V= R^3; T(x1,x2,x3) = (x1+x2, 2x2+2x3, 3x3)

Let V be a three-dimensional vector space with ordered basis B =
{u, v, w}.
Suppose that T is a linear transformation from V to itself and
T(u) = u + v,
T(v) = u, T(w) =
v.
1. Find the matrix of T relative to the ordered basis B.
2. A typical element of V looks like
au + bv +
cw, where a, b and c
are scalars. Find T(au +
bv + cw). Now
that you know...

Suppose V is a vector space and T is a linear operator. Prove by
induction that for all natural numbers n, if c is an eigenvalue of
T then c^n is an eigenvalue of T^n.

Find the coordinates of e1 e2
e3 of R3 in terms of [(1,0,0)T ,
(1,1,0)T , (1,1,1)T ] of R3,, and
then find the matrix of the linear transformation T(x1,,
x2 , x3 )T = [(4xx+
x2- x3)T , (x1 +
3x3)T , (x2 +
2x3)T with respect to this basis.

10 Linear Transformations. Let V = R2 and W = R3. Define T: V →
W by T(x1, x2) = (x1 − x2, x1, x2). Find the matrix representation
of T using the standard bases in both V and W
11 Let T :R3 →R3 be a linear transformation such that T(1, 0, 0)
= (2, 4, −1), T(0, 1, 0) = (1, 3, −2), T(0, 0, 1) = (0, −2, 2).
Compute T(−2, 4, −1).

Let
V be a vector space with ordered basis B = (b1, . . . , bn). Does
the basis having n elements imply that V is the coordinate space
R^n?

Consider Linear transformation in P2(R)
T(x)=((1-x2)f'(x))'. Compose a basis for P2(R) composed
of eigenvectors of T.

(a) Prove that if two linear transformations T,U : V --> W
have the same values on a basis for V, i.e., T(x) = U(x) for all x
belong to beta , then T = U. Conclude that every linear
transformation is uniquely determined by the images of basis
vectors.
(b) (7 points) Determine the linear transformation T : P1(R)
--> P2(R) given by T (1 + x) = 1+x^2, T(1- x) = x by finding the
image T(a+bx) of...

Consider the ordered bases B={[8,9]} and
C={[-2,0],[-3,3]} for the vector space R^2.
A. find the matrix from C to B.
B.Find the coordinates of u=[2,1] in the ordered basis B.
C.Find the coordinates of v in the ordered basis B if the
coordinate vector of v in C =[-1,2].

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