Question

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

Answer #1

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...

Suppose T : V → W is a homomorphism.
Prove:
(i) If dim(V ) < dim(W) then T is not surjective.
(ii) If dim(V ) > dim(W) then T is not injective.

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

Let n be in N and let K be a field. Show that for a linear map T
: Kn to Kn the following statements are
equivalent:
1. The map T is one-to-one (injective).
2. The map T is onto (surjective).
3. The map T is invertible.
4. The map T is an isomorphism.

Problem 3. Recall that a linear map f : V → W is called an
isomorphism if it is invertible (i.e. has a linear inverse map). We
proved in class that f is in fact invertible if and only if it is
bijective. Use this fact from class together with the Rank-Nullity
Theorem (of the previous problem) to show that if f : V → V is an
endomorphism, then it is actually invertible if
1. it is merely injective...

Let (V, |· |v ) and (W, |· |w ) be normed vector spaces. Let T :
V → W be linear map. The kernel of T, denoted ker(T), is defined to
be the set ker(T) = {v ∈ V : T(v) = 0}. Then ker(T) is a linear
subspace of V .
Let W be a closed subspace of V with W not equal to V . Prove
that W is nowhere dense in V .

1.4.3. Let T(x) =Ax+b be an invertible affine transformation of
R3.
Show that T ^-1 is also affine.

Let V be a finite-dimensional vector space over C and T in L(V)
be an invertible operator in V. Suppose also that T=SR is the polar
decomposition of T where S is the correspondIng isometry and
R=(T*T)^1/2 is the unique positive square root of T*T. Prove that R
is an invertible operator that committees with T, that is
TR-RT.

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 W be an inner product space and v1,...,vn a basis of V. Show
that〈S, T 〉 = 〈Sv1, T v1〉 + . . . + 〈Svn, T vn〉
for S,T ∈ L(V,W) is an inner product on L(V,W).
Let S ∈ L(R^2) be given by S(x1, x2) = (x1 + x2, x2) and let I ∈
L(R^2) be the identity operator. Using the inner product defined in
problem 1 for the standard basis and the dot product, compute 〈S,...

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