Question

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.

Answer #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

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 (u,v,w,t) be a linearly independent list of vectors in R4.
Determine if (u, v-u, w+5v, t) is a linearly independent list.
Explain your reasoning and Show work.

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

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.

(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 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 u=(-1, 2, 4)T and v=(4, a, 1)T . For
what value of "a" are these vectors linearly
independent?
Insert the value of "a".

(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...

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