Consider the linear transformation T: R2x2 ->
R2x2 defined by T(A) = AT - A.
Determine the eigenvalues of this linear transformation and
their algebraic and geometric multiplicities.
For each statement, determine whether the statement is true or false. Give a sentence justifying your answer.
a. If tan(t)=0, then cos(t)=1 .
b. The function f(x)=sin(x)cos(x) has period 2π.
c. The graph of r = 1 is the unit circle.
d. Two angles with the same cosine value must have the same sine value.
e. The point (0, -3) in Cartesean coordinates can also be described by the ordered pair (-3, π/2) in polar coordinates
Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) =
(2,3,-5) and T( e⃗2 ) = (-1,0,1).
Determine the standard matrix of T.
Calculate T( ⃗u ), the image of ⃗u=(4,2) under T.
Suppose T(v⃗)=(3,2,2) for a certain v⃗ in R2 .Calculate the
image of ⃗w=2⃗u−v⃗ .
4. Find a vector v⃗ inR2 that is mapped to ⃗0 in R3.
. Let T : R n → R m be a linear transformation and A the
standard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basis
for ker(T) (i.e. Null(A)). We extend BN to a basis for R n and
denote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show the
set BR = {T( r~u +1), . . . , T( ~un)} is a...
Let T: R2 -> R2 be a linear
transformation defined by T(x1 , x2) =
(x1 + 2x2 , 2x1 +
4x2)
a. Find the standard matrix of T.
b. Find the ker(T) and nullity (T).
c. Is T one-to-one? Explain.
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?
The linear transformation is such that for any v in
R2, T(v) = Av.
a) Use this relation to find the image of the vectors
v1 = [-3,2]T and v2 =
[2,3]T. For the following transformations take k = 0.5
first then k = 3,
T1(x,y) = (kx,y)
T2(x,y) = (x,ky)
T3(x,y) = (x+ky,y)
T4(x,y) = (x,kx+y)
For T5 take theta = (pi/4) and then theta =
(pi/2)
T5(x,y) = (cos(theta)x - sin(theta)y, sin(theta)x +
cos(theta)y)
b) Plot v1 and...
Define the linear transformation S : Pn →
Pn and T : Pn → Pn by S(p(x)) =
p(x + 1), T(p(x)) = p'(x)
(a) Find the matrix associated with S and T with respect to the
standard basis {1, x, x2} for P2 .
(b) Find the matrix associated with S ◦ T(p(x)) for n = 2 and
for the standard basis {1, x, x2}. Is the linear
transformation S ◦ T invertible?
(c) Is S a one-to-one transformation?...