Question

(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 a+bx for a+bx belong to P1(R). Here, {1+x,1 x} is a basis for P1(R).

(c) (3 points) Is the linear map T from part b. injective? Explain your answer.

Answer #1

Let T:V→W be a linear transformation and U be a subspace of V.
Let T(U)T(U) denote the image of U under T (i.e., T(U)={T(u⃗ ):u⃗
∈U}). Prove that T(U) is a subspace of W

Suppose that V is a vector space with basis {u,
v, w}. Suppose that T is a linear
transformation from V to W and suppose also that
{T(u), T(v),
T(w)} is a basis for W. Prove from the
definitions that T is both 1-1 and onto.

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: U--> V be a linear transformation. Prove that the
range of T is a subspace of W

Let T ∈ L(U,V) and S ∈ L(V,W). Prove that the product ST is a
linear map and hence ST ∈ L(U,W). Please be specific about each
step, thank you!

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 T be a linear transformation that is one-to-one, and u, v be
two vectors that are linearly independent. Is it true that the
image vectors T(u), T(v) are linearly independent? Explain why or
why not.

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

1. Let T be a linear transformation from vector spaces
V to W.
a. Suppose that U is a subspace of V,
and let T(U) be the set of all vectors w in W
such that T(v) = w for some v in V. Show that
T(U) is a subspace of W.
b. Suppose that dimension of U is n. Show that
the dimension of T(U) is less than or equal to
n.

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...

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