ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry Flashcards

24 sample cards from the Mathematics (Part I) - Algebra and Trigonometry deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define a relation from set A to set B.

   A relation from A to B is any subset of the Cartesian product A × B (a set of ordered pairs).

2. Define the domain and range of a relation.

   The domain is the set of all first elements of the ordered pairs; the range is the set of all second elements.

3. Define a function from set A to set B.

   A function f: A → B assigns to every element of A exactly one element of B (no element of A has two images and no element is left unmapped).

4. Distinguish between an into function and an onto (surjective) function.

   In an into function, at least one element of the codomain has no pre-image (range ⊂ codomain). In an onto function the range equals the codomain (every element of B is an image).

5. Define a one-to-one (injective) function and a bijective function.

   Injective: distinct elements of the domain have distinct images (f(a₁)=f(a₂) ⇒ a₁=a₂). Bijective: a function that is both one-to-one and onto.

6. Define a binary operation on a set S.

   A binary operation * on S is a function that assigns to each ordered pair (a, b) of elements of S a unique element a*b that is also in S (closure).

7. List the four axioms a set with a binary operation must satisfy to be a group.

   1) Closure, 2) Associativity, 3) Existence of an identity element e, 4) Existence of an inverse for every element. (An abelian group additionally satisfies commutativity.)

8. What is the difference between a semigroup, a monoid, and a group?

   Semigroup: closure + associativity. Monoid: semigroup + identity element. Group: monoid + inverse for every element.

9. Define an abelian (commutative) group.

   A group (G, *) in which the operation is commutative, i.e. a*b = b*a for all a, b ∈ G.

10. Define a matrix and its order (dimension).

    A matrix is a rectangular array of numbers arranged in rows and columns. Its order is m × n where m is the number of rows and n the number of columns.

11. Define a square matrix, a row matrix and a column matrix.

    Square matrix: equal number of rows and columns (n × n). Row matrix: a single row (1 × n). Column matrix: a single column (m × 1).

12. Define a diagonal matrix, a scalar matrix and an identity matrix.

    Diagonal: square matrix with all off-diagonal entries zero. Scalar: diagonal matrix with all diagonal entries equal. Identity: scalar matrix with all diagonal entries 1.

13. Define the transpose of a matrix, and what is a symmetric matrix?

    The transpose Aᵗ is obtained by interchanging rows and columns. A is symmetric if Aᵗ = A; it is skew-symmetric if Aᵗ = −A.

14. State the condition for two matrices to be conformable for multiplication and the order of the product.

    A (m×n) and B (p×q) can be multiplied only if n = p (columns of A = rows of B); the product AB has order m × q.

15. Is matrix multiplication commutative? State the correct general property.

    No, matrix multiplication is generally non-commutative: AB ≠ BA in general. However it is associative ((AB)C = A(BC)) and distributive over addition.

16. State the transpose properties (AB)ᵗ and (A+B)ᵗ.

    (AB)ᵗ = Bᵗ Aᵗ (reversal law) and (A + B)ᵗ = Aᵗ + Bᵗ. Also (Aᵗ)ᵗ = A.

17. How do you evaluate the determinant of a 2×2 matrix [[a, b],[c, d]]?

    det = ad − bc.

18. What is a singular matrix and a non-singular matrix?

    A square matrix is singular if its determinant is 0; it is non-singular (invertible) if its determinant is non-zero.

19. State the determinant properties for det(Aᵗ) and det(AB).

    det(Aᵗ) = det(A) and det(AB) = det(A) · det(B).

20. What happens to a determinant if two rows (or columns) are interchanged, or if a row is multiplied by a scalar k?

    Interchanging two rows/columns changes the sign of the determinant. Multiplying one row/column by k multiplies the whole determinant by k. If two rows/columns are identical, the determinant is 0.

21. Give the formula for the inverse of a non-singular matrix A.

    A⁻¹ = (1/det A) · adj(A), where adj(A) is the adjoint (transpose of the matrix of cofactors). This requires det A ≠ 0.

22. How do you find the inverse of a 2×2 matrix A = [[a, b],[c, d]]?

    A⁻¹ = 1/(ad − bc) · [[d, −b],[−c, a]], provided ad − bc ≠ 0.

23. Using matrices, how is a system AX = B solved when A is non-singular?

    By the matrix inverse method: X = A⁻¹B, valid when det(A) ≠ 0 (the system has a unique solution).

24. State Cramer's Rule for solving a system of linear equations.

    For AX = B with det(A)=D≠0, each variable xᵢ = Dᵢ / D, where Dᵢ is the determinant of A with its i-th column replaced by the column B.

What this deck covers

Mathematics (Part I) - Algebra and Trigonometry flashcards FAQ