Question

Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be a set of inner products on V that is (when viewed as a subset of B(V)) linearly independent. Prove that S must be finite

e) Exhibit an infinite linearly independent set of inner products on R(x), the vector space of all polynomials with real coefficients.

Answer #1

d) Since V is finite dimensional implies , the dual of V is also finite dimensional and since S be a subset of , which is linearly independent implies S is a linearly independent subset of a finite dimensional vector space . Hence finite.

e) Consider the set . We claim that the given set is linearly independent. Note that if the set is linearly dependent then there exists a natural number n real numbers , i=0,...,n, such that , then consider the polynomial , this this polynomial has more than n roots. Which contradicts the fundamental theorem of algebra. Hence the given set is linearly independent.

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 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 V be a finite dimensional vector space over R with an inner
product 〈x, y〉 ∈ R for x, y ∈ V .
(a) (3points) Let λ∈R with λ>0. Show that
〈x,y〉′ = λ〈x,y〉, for x,y ∈ V,
(b) (2 points) Let T : V → V be a linear operator, such that
〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V.
Show that T is one-to-one.
(c) (2 points) Recall that the norm of a vector x ∈ V...

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}...

Suppose that V is a finite dimensional inner product space over
C and dim V = n, let T be a normal linear transformation of V
If S is a linear transformation of V and T has n distinc
eigenvalues such that ST=TS. Prove S is normal.

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...

(3) Let V be a finite dimensional vector space, and let T: V® V
be a linear transformation such that rk(T) = rk(T2).
a) Show that ker(T) = ker(T2).
b) Show that 0 is the only vector that lies in
both the null space of T, and the range space of T

Let B be a (finite) basis for a vector space V. Suppose that
v is a vector in V but not in B. Prove that, if we
enlarge B by adding v to it, we get a set that
cannot possibly be a basis for V. (We have not yet formally defined
dimension, so don't use that idea in your proof.)

Let S={u,v,w}S={u,v,w} be a linearly independent set in a vector
space V. Prove that the set S′={3u−w,v+w,−2w}S′={3u−w,v+w,−2w} is
also a linearly independent set in V.

1.
Suppose that ? is a finite dimensional vector space over R. Show
that if ???(? ) is odd, then every ? ∈ L(? ) has an eigenvalue.
(Hint: use induction).
(please provide a detailed proof)
2. Suppose that ? is a finite dimensional vector space over R
and ? ∈ L(? ) has no eigenvalues. Prove that every ? -invariant
subspace of ? has even dimension.

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