Question

Suppose V is a vector space and T is a linear operator. Prove by
induction that for all natural numbers n, if c is an eigenvalue of
T then c^n is an eigenvalue of T^n.

Answer #1

1. Assume that V is a vector space and L is a linear function V
→ V.
a. Suppose there are two vectors v and w in V such that v, w,
and v+w are all eigenvectors of L. Show that v and w share the same
eigenvalue.
b. Suppose that every vector in V is an eigenvector of L. Prove
that there is a scalar α such that L = αI.

Let V be a finite-dimensional vector space and let T be a linear
map in L(V, V ). Suppose that dim(range(T 2 )) = dim(range(T)).
Prove that the range and null space of T have only the zero vector
in common

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

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.

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

Suppose V is a ﬁnite dimensional inner product space. Prove that
every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V
, is an isomorphism.

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

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 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.

5. Let V be a finite-dimension vector space and T : V → V be
linear. Show that V = im(T) + ker(T) if and only if im(T) ∩ ker(T)
= {0}.

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