Question

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).

Answer #1

Let V = W = R2. Choose the basis B = {x1, x2} of V , where x1 =
(2, 3), x2 = (4,−5) and choose the basis D = {y1,y2} of W, where y1
= (1,1), y2 = (−3,4). Find the matrix of the identity linear
mapping I : V → W with respect to these bases.

(a) Let T be any linear transformation from R2 to
R2 and v be any vector in R2 such that T(2v)
= T(3v) = 0. Determine whether the following is true or false, and
explain why: (i) v = 0, (ii) T(v) = 0.
(b) Find the matrix associated to the geometric transformation
on R2 that first reflects over the y-axis and then
contracts in the y-direction by a factor of 1/3 and expands in the
x direction by a...

Let T ∈ L(R2) be the linear transformation T(x1, x2) = (3x1 +
2x2, −4x1 − 3x2), v = (1, −1), and p(z) = z^2 − 3z + 2. Compute
p(T), show that p(T)v = 0, and show that NOT all the roots of p(z)
are eigenvalues of T.

(12) (after 3.3)
(a) Find a linear transformation T : R2 → R2 such that T (x) =
Ax that reflects a
vector (x1, x2) about the x2-axis.
(b) Find a linear transformation S : R2 → R2 such that T(x) =
Bx that rotates a
vector (x1, x2) counterclockwise by 135 degrees.
(c) Find a linear transformation (with domain and codomain)
that has the effect
of first reflecting as in (a) and then rotating as in (b).
Give the...

Let the linear transformation T: V--->W be such that T (u) =
u2 If a, b are Real. Find T (au + bv) ,
if u = (x, y) v = (z, w) and uv = (xz-yw, xw + yz)
Let the linear transformation T: V---> W be such that T (u)
= T (x, y) = (xy, 0) where u = (x, y), with 2, -3. Then, if u = (
1.0) and v = (0.1). Find the value...

Let T be the function from R2 to R3 defined by T ( (x,y) ) = (x,
y, 0). Prove that T is a linear transformation, that it is 1-1, but
that it is not onto.

Let T be the linear transformation from R2 to R2, that rotates a
vector clockwise by 60◦ about the origin, then reﬂects it about the
line y = x, and then reﬂects it about the x-axis.
a) Find the standard matrix of the linear transformation T.
b) Determine if the transformation T is invertible. Give detailed
explanation. If T is invertible, ﬁnd the standard matrix of the
inverse transformation T−1.
Please show all steps clearly so I can follow your...

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.

(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...

let
T: V ->W be a linear transformation. Show that if T is an
isophormism and B is a basis of V, then T(B) is a basis of W

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