- If A,B, and C are each n × n matrices and ABC is invertible, why are each of A,B,
and C invertible?
- Give an example of a 3 × 2 matrix with the property that the linear transformation
determined by this matrix is one to one but not onto.
- Explain why Ax = 0 always has a solution whenever A is a linear transformation.
- Review problem: Suppose det = 0
. Show using Theorem 3.1.15 there exists
x≠0 such that
x = 0.
- How does the minimal polynomial of an algebraic number relate to the minimal
polynomial of a linear transformation? Can an algebraic number be thought of as a
linear transformation? How?
- Recall the fact from algebra that if p and
q are polynomials, then there exists
polynomial such that
where the degree of r is less than the degree of
p or else
r = 0
this in mind, why must the minimal polynomial always divide the characteristic
polynomial? That is, why does there always exist a polynomial l such that
q? Can you give conditions which imply the minimal polynomial
equals the characteristic polynomial? Go ahead and use the Cayley Hamilton
- In the following examples, a linear transformation, T is given by specifying its action on a
basis β. Find its matrix with respect to this basis.
- T = 2
- T = 2
- T = 2
,T = 1
- Let β = be a basis for
Fn and let T : Fn → Fn be defined as follows.
First show that T is a linear transformation. Next show that the matrix of T with respect to
Show that the above definition is equivalent to simply specifying T on the basis vectors of β
- ↑In the situation of the above problem, let γ = be the standard basis for
where ek is the vector which has 1 in the kth entry and zeros elsewhere. Show that
- ↑Generalize the above problem to the situation where T is given by specifying its action on
the vectors of a basis β = as follows.
Letting A =
, verify that for γ =
, 8.5 still holds and that
β = A.
- Let P3 denote the set of real polynomials of degree no more than 3, defined on an interval
. Show that
P3 is a subspace of the vector space of all functions defined on this interval.
Show that a basis for P3 is
. Now let D denote the differentiation operator
which sends a function to its derivative. Show D is a linear transformation which sends P3
to P3. Find the matrix of this linear transformation with respect to the given
- Generalize the above problem to Pn, the space of polynomials of degree no more than n with
- In the situation of the above problem, let the linear transformation be T = D2 + 1, defined
as Tf = f′′ + f. Find the matrix of this linear transformation with respect to the given basis
. Write it down for
n = 4.
- In calculus, the following situation is encountered. There exists a vector valued function
f :U → ℝm where U is an open subset of ℝn. Such a function is said to have a derivative or
to be differentiable at x ∈ U if there exists a linear transformation T : ℝn → ℝm such
First show that this linear transformation, if it exists, must be unique. Next show that for
,, the standard basis, the kth column of
Actually, the result of this problem is a well kept secret. People typically don’t see this in
calculus. It is seen for the first time in advanced calculus if then.
- Recall that A is similar to B if there exists a matrix P such that A = P−1BP. Show that if
A and B are similar, then they have the same determinant. Give an example of two matrices
which are not similar but have the same determinant.
- Suppose A ∈ℒ where dim
. Show ker
≠. That is, show there
exist nonzero vectors
v ∈ V such that Av = 0.
- A vector v is in the convex hull of a nonempty set S if there are finitely many vectors of
S, and nonnegative scalars
Such a linear combination is called a convex combination. Suppose now that S ⊆ V, a vector
space of dimension n. Show that if v =∑
k=1mtkvk is a vector in the convex hull for
m > n + 1, then there exist other scalars such that
Thus every vector in the convex hull of S can be obtained as a convex combination of at
most n + 1 points of S. This incredible result is in Rudin . Hint: Consider
L : ℝm → V × ℝ defined by
Explain why ker
. Next, letting a ∈ ker
λ ∈ ℝ, note that
. Thus for all λ ∈ ℝ,
Now vary λ till some tk + λak = 0 for some ak≠0.
- For those who know about compactness, use Problem 17 to show that if S ⊆ ℝn and S is
compact, then so is its convex hull.
- Suppose Ax = b has a solution. Explain why the solution is unique precisely when Ax = 0
has only the trivial (zero) solution.
- Let A be an n×n matrix of elements of F. There are two cases. In the first case, F contains
a splitting field of pA so that
p factors into a product of linear polynomials having
F. It is the second case which is of interest here where pA does not factor
into linear factors having coefficients in
F. Let G be a splitting field of pA and let
qA be the minimal polynomial of
A with respect to the field G. Explain why
qA must divide
pA. Now why must
qA factor completely into linear
- In Lemma 8.2.2 verify that L is linear.