BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 5

Question 1

What is the definition of a bijective function?

Accepted Answer

A function that is both injective (one-to-one) and surjective (onto)

Answer

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?

Accepted Answer

A commutative ring in which every nonzero element has a multiplicative inverse

Answer

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:

Accepted Answer

A normal distribution as the sample size increases, regardless of the population's distribution

Answer

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?

Accepted Answer

The matrix A is square and its determinant is nonzero

Answer

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?

Accepted Answer

All of ℝ²

Answer

A 2×3 matrix of rank 2 has image of dimension 2, which spans all of ℝ² (the codomain has dimension 2).

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Bachelor of Mathematics: Concepts and Theories 5