Algebra II 189-251B Instructor: Dr. E. Goren.

Midterm Exam. February 26, 2001.

Calculators, books or notes are not allowed. Answer all questions in Part A and B. Part C is optional.

Notation. The symbols ℤ, ℝ, stand for the integers and real numbers, respectively. If 𝔽 is a field we denote by 𝔽[t]_{n} the vector space over 𝔽 of polynomials with coefficients in 𝔽 and of degree less than n.

PART A. Each question has a unique correct answer. Give only that answer number with no explanations or proofs. Each question is worth 8 points.

1. There exists a surjective linear transformation from ℝ^{5} to ℝ[t]_{6}.

1. Yes.

2. No.

2. Let 𝔽 be the field ℤ/7ℤ. There exists a non-zero polynomial f ∈ 𝔽[t]_{4} such that

f(0) = 0, f(1) + f(3) = 0, f′(0) = 0.

1. Yes.

2. No.

3. Let A be an invertible n × n matrix with entries in a field 𝔽. Let St be the standard basis for 𝔽^{n}. There exists a basis B such that A = _{St}M_{B}.

1. Yes.

2. No.

4. The rank of the matrix (\begin{matrix}1 & 3 & 6 \\ 1 & 2 & 3 \\ 1 & 5 & 0 \end{matrix}) over the field ℤ/3ℤ is

1. 0.

2. 1.

3. 2.

4. 3.

5. Let B = {(12, 13), (−11, 3)} be a basis of ℝ^{2}. There exists a linear transformation T : ℝ^{2} ⟶ ℝ^{2} such that

St[T]_{St} = (\begin{matrix}1 & 3 \\ 0 & 0 \end{matrix}) , _{B}[T]_{B} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) .

1. Yes.

2. No.

PART B. Answer all questions. Each question is worth 30 points.

1. (A) Prove that a maximal independent set is a spanning set (no need to prove it is minimal). (B) Prove that for any two n × n matrices A, B we have

det(AB) = det(A) det(B).

2. Let W be the subspace of ℝ^{4} consisting of the solutions to the equation

x_{1} + x_{2} + x_{3} − x_{4} = 0.

Let U be the subspace of ℝ^{4} equal to the span of the vector (1, 1, 1, −1).

Find the projection on W along U (write it as a matrix with respect to the standard basis). What is the image of the vector (1, −1, 1, −1) under this projection?

PART C. This part is optional and is worth 15 points.

Let V be a vector space over a field 𝔽, perhaps of infinite dimension. We say that a linear operator T : V ⟶ V is nilpotent if for every v ∈ V there exists an integer n(v) ≥ 1 such that T^{n(v)}(v) = 0. Show

1. If V has finite dimension this is equivalent to the definition of a nilpotent operator that we gave, namely, there exists an integer N ≥ 1 such that T^{N} ≡ 0.

2. Give an example showing Fitting’s lemma fails for infinite dimensional vector spaces.