Question

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 T . When T has an inverse, we say ? is invertible.

Show that

(a) T has a right inverse *iff* T is
surjective.

(b) If ? is a basis for ? and ? is a basis for ?, then [?]?? has
a right inverse *iff* its rows are linearly independent.

Answer #1

let T:V to W be a linear transdormation of vector
space V and W and let B=(v1,v2,...,vn) be a basis for V. Show that
if (Tv1,Tv2,...,Tvn) is linearly independent, thenT is
injecfive.

Construct a linear transformation T : V → W, where V and W are
vector spaces over F such that the dimension of the kernel space of
T is 666. Is such a transformation unique?
Give reasons for your answer.

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 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 L : V → W be a linear transformation between two vector
spaces. Show that dim(ker(L)) + dim(Im(L)) = dim(V)

Let V and W be finite-dimensional vector spaces over F, and let
φ : V → W be a linear transformation. Let dim(ker(φ)) = k, dim(V )
= n, and 0 < k < n. A basis of ker(φ), {v1, . . . , vk}, can
be extended to a basis of V , {v1, . . . , vk, vk+1, . . . , vn},
for some vectors vk+1, . . . , vn ∈ V . Prove that...

Let T:V-->V be a linear transformation and let T^3(x)=0 for
all x in V. Prove that R(T^2) is a subset of N(T).

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 β={v1,v2,…,vn} be a basis for a vector space V
and T:V→V be a linear transformation. Prove that [T]β is upper
triangular if and only if T(vj)∈span({v1,v2,…,vj}) j=1,2,…,n. Visit
goo.gl/k9ZrQb for a solution.

a. Let T : V → W be left invertible. Show that T is
injective.
b. Let T : V → W be right invertible. Show that T is
surjective

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