BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 5 — Questions and Answers
Question 1: What is the definition of a bijective function?
- A function that is both injective (one-to-one) and surjective (onto) (Correct answer)
- A function whose codomain equals its domain
- A function with a continuous inverse
- A function that maps distinct inputs to distinct outputs only
Correct answer: A function that is both injective (one-to-one) and surjective (onto)
A bijection is simultaneously injective (no two inputs share an output) and surjective (every output is achieved), making it invertible.
Question 2: Which of the following best describes the concept of a field in abstract algebra?
- A commutative ring in which every nonzero element has a multiplicative inverse (Correct answer)
- A group with two binary operations
- A ring in which every ideal is principal
- A set closed under addition and scalar multiplication
Correct answer: A commutative ring in which every nonzero element has a multiplicative inverse
A field is a commutative ring where the nonzero elements form a group under multiplication, allowing division by any nonzero element.
Question 3: In statistics, the Central Limit Theorem states that the sampling distribution of the sample mean approaches:
- A normal distribution as the sample size increases, regardless of the population's distribution (Correct answer)
- A uniform distribution for large samples
- The same distribution as the population
- A t-distribution for all sample sizes
Correct answer: A normal distribution as the sample size increases, regardless of the population's distribution
The CLT guarantees that with sufficient sample size, the distribution of sample means is approximately normal even if the underlying population is not.
Question 4: Which condition ensures that a system of linear equations Ax = b has a unique solution?
- The matrix A is square and its determinant is nonzero (Correct answer)
- The matrix A has more rows than columns
- The vector b is in the column space of A
- The matrix A is symmetric
Correct answer: The matrix A is square and its determinant is nonzero
A unique solution exists if and only if A is invertible, which for square matrices is equivalent to det(A) ≠ 0.
Question 5: What is the image (range) of the linear transformation T: ℝ³ → ℝ² defined by a 2×3 matrix of rank 2?
- All of ℝ² (Correct answer)
- A 1-dimensional subspace of ℝ²
- A 2-dimensional subspace of ℝ³
- A single point in ℝ²
Correct answer: All of ℝ²
A 2×3 matrix of rank 2 has image of dimension 2, which spans all of ℝ² (the codomain has dimension 2).
Question 6: Which of the following inequalities is known as the Cauchy-Schwarz inequality in an inner product space?
- |⟨u, v⟩|² ≤ ⟨u, u⟩ · ⟨v, v⟩ (Correct answer)
- |⟨u, v⟩| ≤ ‖u‖ + ‖v‖
- ‖u + v‖ ≤ ‖u‖ + ‖v‖
- ⟨u, v⟩ ≥ 0 for all u, v
Correct answer: |⟨u, v⟩|² ≤ ⟨u, u⟩ · ⟨v, v⟩
The Cauchy-Schwarz inequality bounds the squared inner product by the product of the squared norms, with equality iff u and v are linearly dependent.
Question 7: A function f: ℝ → ℝ is uniformly continuous if:
- For every ε > 0 there exists δ > 0 such that |x − y| < δ implies |f(x) − f(y)| < ε for all x, y (Correct answer)
- For every x and every ε > 0 there exists δ > 0 (depending on x) such that |x − y| < δ implies |f(x) − f(y)| < ε
- f is differentiable everywhere with bounded derivative
- f is continuous and its domain is compact
Correct answer: For every ε > 0 there exists δ > 0 such that |x − y| < δ implies |f(x) − f(y)| < ε for all x, y
Uniform continuity requires a single δ that works for all points simultaneously, unlike pointwise continuity where δ may depend on the point.
BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 5
What is the definition of a bijective function?