Question

- Let the linear transformation T: V--->W be such that T (u) =
u
^{2}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 of
**T (2u + (- 3) v)** - Let the linear transformation T: V----> W be such that T (u)
= T (x, y) = (0, x + y) where u = (x, y), with -7, 8 .
**Find the value of T (au + bv)**if u = (1.0) and v = (0.1)

Answer #1

Given the set S = {(u,v): 0<= u<=4 and 0<= v<=3} and
the transformation T(u, v) = (x(u, v), y(u, v)) where x(u, v) = 4u
+ 5v and y(u, v) = 2u -3v,
graph the image R of S under the transformation T in the
xy-plan
and find the area of region R

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...

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

Let T: U--> V be a linear transformation. Prove that the
range of T is a subspace of 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 and W be vector spaces and let T:V→W be a linear
transformation. We say a linear transformation S:W→V is a left
inverse of T if ST=Iv, where ?v denotes the identity transformation
on V. We say a linear transformation S:W→V is a right inverse of ?
if ??=?w, where ?w denotes the identity transformation on W.
Finally, we say a linear transformation S:W→V is an inverse of ? if
it is both a left and right inverse 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

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).

1) Let T : V —> W be a linear transformation with dim(V) = m
and dim(W) = n. For which of the following conditions is T
one-to-one?
(A) m>n
(B) range(T)=Wandm=n
(C) nullity(T) = m
(D) rank(T)=m-1andn>m
2) For which of the linear transformations is nullity (T) = 0 ?
Why?
(A)T:R3 —>R8 (B)T:P3 —>P3 (C)T:M23 —>M33 (D)T:R5
—>R2
withrank(T)=2 withrank(T)=3 withrank(T)=6 withrank(T)=1

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

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