- 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 thatx = 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 pand qare polynomials, then there exists l, a polynomial such that
where the degree of r

is less than the degree of por else r= 0 . With this in mind, why must the minimal polynomial always divide the characteristic polynomial? That is, why does there always exist a polynomial lsuch that pl= q? Can you give conditions which imply the minimal polynomial equals the characteristic polynomial? Go ahead and use the Cayley Hamilton theorem. - 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+ 1,T=
- T= 2+ 1,T=
- T= 2+ 1,T= 1−

- T
- Let β = be a basis for F
^{n}and let T : F^{n}→ F^{n}be defined as follows.First show that T is a linear transformation. Next show that the matrix of T with respect to this basis,

_{β}isShow that the above definition is equivalent to simply specifying T on the basis vectors of β by

- ↑In the situation of the above problem, let γ = be the standard basis for F
^{n}where e_{k}is the vector which has 1 in the k^{th}entry and zeros elsewhere. Show that_{γ}=(8.5) - ↑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 P
_{3}denote the set of real polynomials of degree no more than 3, defined on an interval. Show that P_{3}is a subspace of the vector space of all functions defined on this interval. Show that a basis for P_{3}is. Now let D denote the differentiation operator which sends a function to its derivative. Show D is a linear transformation which sends P_{3}to P_{3}. Find the matrix of this linear transformation with respect to the given basis. - Generalize the above problem to P
_{n}, the space of polynomials of degree no more than n with basis. - In the situation of the above problem, let the linear transformation be T = D
^{2}+ 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 → ℝ
^{m}where U is an open subset of ℝ^{n}. Such a function is said to have a derivative or to be differentiable at x ∈ U if there exists a linear transformation T : ℝ^{n}→ ℝ^{m}such thatFirst show that this linear transformation, if it exists, must be unique. Next show that for β =

,, the standard basis, the k^{th}column of_{β}isActually, 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
^{−1}BP. 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 ∈ℒwhere dim> 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 scalarssuch that
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=1}^{m}t_{k}v_{k}is a vector in the convex hull for m > n + 1, then there exist other scalarssuch thatThus 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 [24]. Hint: Consider L : ℝ

^{m}→ V × ℝ defined byExplain why ker

≠. Next, letting a ∈ ker∖and λ ∈ ℝ, note that λa ∈ker. Thus for all λ ∈ ℝ,Now vary λ till some t

_{k}+ λa_{k}= 0 for some a_{k}≠0. - For those who know about compactness, use Problem 17 to show that if S ⊆ ℝ
^{n}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 p
_{A}so that pfactors into a product of linear polynomials having coefficients in F. It is the second case which is of interest here where p_{A}does not factor into linear factors having coefficients in F. Let G be a splitting field of p_{A}and let q_{A}be the minimal polynomial of A with respect to the field G. Explain why q_{A}must divide p_{A}. Now why must q_{A}factor completely into linear factors? - In Lemma 8.2.2 verify that L is linear.

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