Question

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.

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

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}...

Consider vector spaces with scalars in the field F(could be R or
C). Recall that L(V, W) is the vectors space consisting of all
linear transformations from V to W.
a. Prove that L(F, W) is isomorphic to W.
b. Assume that V is a finite dimensional vectors space. Prove
that L(V, F) is isomorphic to V.
c. If V is infinite dimensional, what happens to L(V, F)?

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.

Linear Algebra Conceptual Questions
• If a subset of a vector space is NOT a subspace, what are the
four things that could go wrong? How could you check to see which
of these four properties aren’t true for the subset?
• Is it possible for two distinct eigenvectors to correspond to
the same eigenvalue?
• Is it possible for two distinct eigenvalues to correspond to
the same eigenvector?
• What is the minimum number of vectors required take to...

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

Suppose V and W are two vector spaces. We can make the set V × W
= {(α, β)|α ∈ V,β ∈ W} into a vector space as follows:
(α1,β1)+(α2,β2)=(α1 + α2,β1 + β2)
c(α1,β1)=(cα1, cβ1)
You can assume the axioms of a vector space hold for V × W
(A) If V and W are finite dimensional, what is the dimension of
V × W? Prove your answer.
Now suppose W1 and W2 are two subspaces of 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.

Let V be an n-dimensional vector space and W a vector
space that is isomorphic to V. Prove that W is also
n-dimensional. Give a clear, detailed, step-by-step
argument using the definitions of "dimension" and "isomorphic"
the Definiton of isomorphic: Let V be an
n-dimensional vector space and W a vector space that is
isomorphic to V. Prove that W is also n-dimensional. Give
a clear, detailed, step-by-step argument using the definitions of
"dimension" and "isomorphic"
The Definition of dimenion: the...

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.

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