251 Matrices and Linear Algebra Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Matrices and linear algebra form one of the most structurally rich topics in the Philippine engineering board exam mathematics coverage. Unlike calculus, which builds on a single variable and moves forward in one direction, linear algebra is inherently multi-dimensional. It deals with systems, structures, and transformations that connect directly to how engineers model and solve real-world problems. Whether you are analyzing a network of electrical circuits, solving a truss system, computing stress distributions, or processing digital signals, linear algebra is the underlying mathematical language.

The PRC licensure examinations test this topic across multiple engineering disciplines. ECE candidates encounter it in signal processing and control systems. EE candidates use it in power system analysis and circuit network equations. CE and ME candidates apply it in structural and mechanical systems. The mathematics section of the board exam tests the pure theory: matrix operations, determinants, systems of equations, and eigenvalue problems while the major subject examinations test the applied forms. Understanding both the vocabulary and the mechanics of this subject is not optional. It is a requirement for passing.

This glossary covers 251 terms drawn from the complete scope of matrices and linear algebra as it appears in Philippine engineering education and licensure review. The entries span matrix types and special properties, arithmetic operations, methods for solving linear systems, determinant theory, vector space concepts, linear transformations, and eigenvalue theory. Each definition is written to be clear, precise, and exam-relevant. You will find the exact language that appears in textbooks, review books, and actual board examination items.

Approach this glossary as a reference and a diagnostic tool. Work through the terms in order to build your vocabulary systematically, then return to specific entries when working through practice problems and you need to confirm a definition. The board exam rewards reviewees who understand not just what a matrix operation produces but also what properties it satisfies, what conditions it requires, and what its result means geometrically or physically. This glossary is designed to build exactly that kind of understanding.

The 251 Matrices and Linear Algebra Terms and Definitions

1. 2 × 2 Determinant

The determinant of a 2 × 2 matrix [[a, b], [c, d]] computed as ad − bc. This is the simplest non-trivial determinant and appears constantly in board exam problems. It represents the signed area of the parallelogram formed by the two row (or column) vectors.

2. 2 × 2 Inverse Formula

The explicit formula for the inverse of a non-singular 2 × 2 matrix. For A = [[a, b], [c, d]] with det(A) = ad − bc ≠ 0, the inverse is A⁻¹ = (1/(ad − bc)) · [[d, −b], [−c, a]]. This formula is one of the most frequently tested computations in the board exam.

3. 3 × 3 Determinant

The determinant of a 3 × 3 matrix, computed using cofactor expansion along any row or column, or using the Rule of Sarrus. The result is a sum of six terms, three positive and three negative, each being a product of three entries with no two from the same row or column.

4. Absolute Value of a Matrix

Not a standard single operation, but the term sometimes refers to taking the absolute value of each individual element of a matrix. In other contexts it refers to the determinant in the sense of magnitude. The most precise usage specifies which interpretation is intended.

5. Addition of Matrices

The operation of adding two matrices of the same dimensions by adding their corresponding elements. If A and B are both m × n matrices, then (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ for all i and j. Matrix addition is commutative and associative.

6. Adjoint Matrix

The transpose of the cofactor matrix of a square matrix, also called the classical adjoint or adjugate. Denoted adj(A). It satisfies the relation A · adj(A) = det(A) · I, which provides the basis for computing the matrix inverse using the formula A⁻¹ = adj(A)/det(A) when det(A) ≠ 0.

7. Adjugate

Another name for the adjoint matrix. The adjugate of a square matrix A is the transpose of its cofactor matrix. It is used in the explicit formula for the matrix inverse and in Cramer’s rule.

8. Affine Combination

A linear combination of vectors in which the coefficients sum to one. Unlike a general linear combination, an affine combination does not require the coefficients to be non-negative. Affine combinations preserve the structure of affine subspaces.

9. Affine Subspace

A subset of a vector space that is a translate of a subspace. It can be written as a particular solution plus the null space of a matrix. The solution set of a consistent non-homogeneous linear system Ax = b is an affine subspace.

10. Algebraic Multiplicity

The number of times a particular eigenvalue appears as a root of the characteristic polynomial of a matrix. An eigenvalue with algebraic multiplicity greater than one is called a repeated eigenvalue. The algebraic multiplicity is always greater than or equal to the geometric multiplicity.

11. Alternating Property of Determinants

The property stating that if any two rows (or two columns) of a square matrix are interchanged, the sign of its determinant reverses. This is one of the three defining properties of the determinant function along with multilinearity and normalization.

12. Associativity of Matrix Multiplication

The property that matrix multiplication is associative: (AB)C = A(BC) for any three matrices with compatible dimensions. Unlike scalar multiplication, this associativity holds for matrices, but commutativity does not.

13. Augmented Matrix

A matrix formed by appending the constant vector b to the coefficient matrix A, written as [A | b]. It is used in Gaussian elimination and Gauss-Jordan elimination to solve systems of linear equations Ax = b by performing row operations on the combined matrix.

14. Back Substitution

The process of solving for unknowns in a system of linear equations after the system has been reduced to upper triangular form by Gaussian elimination. Starting from the last equation, each variable is solved in reverse order by substituting already-known values.

15. Back-Substitution Steps

The sequential process of finding variable values in a system reduced to upper triangular form by starting from the last equation (which has one unknown), solving it, then substituting upward through the remaining equations.

16. Backward Elimination

The portion of the Gauss-Jordan method that eliminates entries above the pivot positions, in contrast to forward elimination which clears entries below. Together, forward and backward elimination reduce the augmented matrix to reduced row echelon form.

17. Basis

A set of linearly independent vectors that spans a vector space. Every vector in the space can be written as a unique linear combination of the basis vectors. The number of vectors in a basis equals the dimension of the vector space, and all bases of the same space have the same number of elements.

18. Basis Vector

Any one of the vectors belonging to a basis of a vector space. In the standard basis of Rⁿ, the basis vectors are the unit coordinate vectors e₁, e₂, …, eₙ, each having a one in one position and zeros elsewhere.

19. Bilinear Form

A function of two vector arguments that is linear in each argument separately. For an n-dimensional real vector space, a bilinear form can be represented as xᵀAy for some matrix A. Symmetric bilinear forms correspond to symmetric matrices.

20. Block Diagonal Matrix

A square matrix composed of square submatrices arranged along its main diagonal, with zero matrices filling all off-diagonal blocks. The determinant of a block diagonal matrix is the product of the determinants of its diagonal blocks. Operations on block diagonal matrices can often be performed block by block.

21. Block Matrix

A matrix partitioned into smaller submatrices called blocks. Block matrix notation allows large matrix equations to be written compactly and computed efficiently by treating blocks as individual elements, provided the dimensions are compatible.

22. Block Matrix Multiplication

The multiplication of two block-partitioned matrices by treating the submatrices as scalars, multiplying and adding them according to the rules of matrix multiplication, provided the dimensions of the blocks are consistent for each product.

23. Cayley-Hamilton Theorem

The theorem stating that every square matrix satisfies its own characteristic equation. If p(λ) = det(λI − A) is the characteristic polynomial of A, then p(A) = 0 (the zero matrix). This theorem is used to compute matrix powers and inverses efficiently.

24. Change of Basis

The process of expressing the coordinates of a vector or the representation of a linear transformation with respect to a new basis. If P is the matrix whose columns are the new basis vectors, then the change of basis transformation is given by the relation x = Px’, where x’ is the coordinate vector in the new basis.

25. Change of Basis Matrix

The invertible matrix P used to convert coordinates from one basis to another. If B and B’ are two bases, the change of basis matrix from B’ to B has the B-coordinates of the B’-basis vectors as its columns. Also called the transition matrix.

26. Characteristic Equation

The polynomial equation det(λI − A) = 0 whose solutions are the eigenvalues of the square matrix A. It is obtained by setting the characteristic polynomial equal to zero. For an n × n matrix, the characteristic equation is a polynomial of degree n in λ.

27. Characteristic Polynomial

The polynomial p(λ) = det(λI − A) associated with a square matrix A. The degree of the characteristic polynomial equals the size of the matrix. Its roots are the eigenvalues of A, and the coefficients encode information about the trace, determinant, and other properties of A.

28. Characteristic Polynomial Coefficients

The coefficients of the characteristic polynomial det(λI − A), which are related to invariants of the matrix. The leading coefficient is always 1, the coefficient of λⁿ⁻¹ is −tr(A), and the constant term is (−1)ⁿ det(A). These relationships are useful for quickly checking calculations.

29. Characteristic Value

Another name for an eigenvalue. The characteristic values of a matrix A are the values of λ that satisfy det(λI − A) = 0. The term is used interchangeably with eigenvalue in many Philippine engineering textbooks.

30. Characteristic Vector

Another name for an eigenvector. A non-zero vector x is a characteristic vector of matrix A corresponding to characteristic value λ if Ax = λx.

31. Cofactor

The signed minor of an element in a square matrix. The cofactor Cᵢⱼ of the element aᵢⱼ is defined as (−1)^(i+j) times the minor Mᵢⱼ, where Mᵢⱼ is the determinant of the submatrix obtained by deleting row i and column j. Cofactors are used in cofactor expansion and in computing the adjoint.

32. Cofactor Expansion

The method of computing the determinant of a square matrix by expanding along any row or column using cofactors. For expansion along row i, det(A) = Σⱼ aᵢⱼ Cᵢⱼ. The result is the same regardless of which row or column is chosen for expansion.

33. Cofactor Matrix

The matrix formed by replacing each element aᵢⱼ of a square matrix with its cofactor Cᵢⱼ. The transpose of the cofactor matrix is the adjoint. The cofactor matrix is an intermediate step in computing the matrix inverse using the adjoint formula.

34. Column Matrix

A matrix with a single column, also called a column vector. It has dimensions n × 1 for some positive integer n. Column matrices represent vectors in Rⁿ and are the standard form for representing unknowns and solutions in linear systems.

35. Column Operations

Elementary operations applied to the columns of a matrix, analogous to elementary row operations. Column operations can be used to simplify matrices or compute determinants but do not preserve the solution set of linear systems (unlike row operations).

36. Column Rank

The maximum number of linearly independent columns in a matrix. The column rank always equals the row rank, and both are equal to the rank of the matrix. This equality is a fundamental theorem of linear algebra.

37. Column Space

The set of all linear combinations of the columns of a matrix A, also called the range or image of A. It is a subspace of Rᵐ for an m × n matrix. The dimension of the column space equals the rank of A.

38. Column Vector

A matrix consisting of a single column with n rows, representing a vector in n-dimensional space. Written as an n × 1 matrix. Most vector operations in linear algebra use column vector notation.

39. Commutative Property

Matrix addition is commutative: A + B = B + A for any two matrices of the same size. However, matrix multiplication is generally not commutative: AB ≠ BA in general. The failure of commutativity under multiplication is one of the most important distinctions between matrix algebra and scalar algebra.

40. Commuting Matrices

Two square matrices A and B that satisfy AB = BA. Most pairs of matrices do not commute. However, diagonal matrices of the same size always commute with each other, and a matrix always commutes with any polynomial expression in itself and with the identity matrix.

41. Compatible Dimensions

The condition required for matrix multiplication to be defined. For the product AB to exist, the number of columns of A must equal the number of rows of B. If A is m × n and B is n × p, then AB is m × p.

42. Complement of a Subspace

A subspace W’ of a vector space V such that every vector in V can be written uniquely as a sum of a vector in W and a vector in W’. If W and W’ are orthogonal complements, the decomposition uses perpendicularity, and the two subspaces together span V with trivial intersection.

43. Complete Solution

The general solution to the linear system Ax = b, written as x = x_p + x_h, where x_p is any particular solution to the non-homogeneous system and x_h is the general solution to the associated homogeneous system Ax = 0. The complete solution describes all possible solutions.

44. Complex Eigenvalue

An eigenvalue of a real matrix that is a complex number with non-zero imaginary part. Complex eigenvalues of real matrices always come in conjugate pairs. They are associated with rotational and oscillatory behavior in linear systems.

45. Complex Matrix

A matrix whose entries are complex numbers. The theory of complex matrices extends real matrix theory using the conjugate transpose (Hermitian transpose) in place of the ordinary transpose. Hermitian matrices are the complex analog of symmetric matrices.

46. Condition Number

A measure of how sensitive the solution of a linear system is to small changes in the input data or coefficient matrix. For an invertible matrix A, the condition number is κ(A) = ‖A‖ · ‖A⁻¹‖. A large condition number indicates a poorly conditioned or nearly singular system.

47. Conformable Matrices

Matrices that satisfy the dimension requirements needed for a specified operation. Two matrices are conformable for addition if they have the same dimensions. Two matrices are conformable for multiplication if the column count of the first equals the row count of the second.

48. Conjugate Transpose

The matrix obtained by taking the transpose of a complex matrix and then taking the complex conjugate of each entry. Denoted A* or A^H (Hermitian transpose). For real matrices, the conjugate transpose equals the ordinary transpose.

49. Consistent System

A system of linear equations that has at least one solution. A system Ax = b is consistent if and only if b lies in the column space of A, or equivalently, if rank([A|b]) = rank(A). A consistent system has either exactly one solution or infinitely many solutions.

50. Coordinate Vector

The vector of coefficients obtained when expressing a given vector as a linear combination of basis vectors. If B = {v₁, v₂, …, vₙ} is a basis and v = c₁v₁ + c₂v₂ + … + cₙvₙ, then [v]_B = (c₁, c₂, …, cₙ) is the coordinate vector of v with respect to B.

51. Cramer’s Rule

A formula for solving a system of n linear equations in n unknowns with a non-singular coefficient matrix A. Each unknown xᵢ is given by det(Aᵢ)/det(A), where Aᵢ is the matrix formed by replacing the ith column of A with the constant vector b. Cramer’s rule is theoretically exact but computationally expensive for large systems.

52. Cross Product

A binary operation on two vectors in three-dimensional space that produces a third vector perpendicular to both. For vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), the cross product u × v can be expressed as the determinant of a 3 × 3 matrix with the standard unit vectors in the first row. The cross product is anti-commutative: u × v = −(v × u).

53. Decomposition

The process of expressing a matrix as a product of simpler matrices that reveal structural properties. Common decompositions include LU decomposition, QR decomposition, eigendecomposition, and singular value decomposition. Each decomposition serves specific computational and analytical purposes.

54. Defective Matrix

A square matrix that does not have a complete set of n linearly independent eigenvectors, making it non-diagonalizable. A defective matrix has at least one eigenvalue whose geometric multiplicity is strictly less than its algebraic multiplicity. Also called a non-diagonalizable matrix.

55. Deficient Matrix

A matrix whose rank is less than the maximum possible value for its dimensions, meaning it has linearly dependent rows or columns. A square deficient matrix is singular and has a zero determinant.

56. Degenerate System

A system of linear equations that has either no solution or infinitely many solutions, as opposed to a system with a unique solution. Degeneracy arises when the coefficient matrix is singular or when the equations are linearly dependent.

57. Dependence Relation

An equation expressing one vector as a linear combination of others, demonstrating that a set of vectors is linearly dependent. If c₁v₁ + c₂v₂ + … + cₖvₖ = 0 with not all cᵢ equal to zero, this is a dependence relation among the vectors.

58. Dependent Equations

Equations in a linear system that are multiples (or linear combinations) of other equations in the system. Dependent equations contribute no new information and result in free variables and infinitely many solutions when the system is consistent.

59. Determinant

A scalar value computed from the elements of a square matrix that encodes important properties including invertibility, volume scaling, and the sign of orientation. The determinant of an n × n matrix is denoted det(A) or |A|. A matrix is invertible if and only if its determinant is non-zero.

60. Diagonal Dominance

A property of a square matrix where the absolute value of each diagonal element is greater than or equal to the sum of the absolute values of all other elements in the same row. Strictly diagonally dominant matrices are non-singular and arise in many numerical methods for solving linear systems.

61. Diagonal Element

An element aᵢᵢ of a matrix located on the main diagonal, where the row index equals the column index. The set of all diagonal elements forms the main diagonal.

62. Diagonal Matrix

A square matrix in which all elements outside the main diagonal are zero. Only the diagonal entries aᵢᵢ may be non-zero. The identity matrix and scalar matrices are special cases of diagonal matrices. Diagonal matrices are easy to multiply: the product of two diagonal matrices is diagonal.

63. Diagonalizable Matrix

A square matrix A that can be written as A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P is an invertible matrix of corresponding eigenvectors. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, where n is its size.

64. Diagonalization

The process of finding matrices P and D such that A = PDP⁻¹, where D is diagonal. The columns of P are the eigenvectors of A and the diagonal entries of D are the corresponding eigenvalues. Diagonalization simplifies the computation of matrix powers and matrix functions.

65. Diagonally Dominant System

A square linear system whose coefficient matrix is diagonally dominant. Such systems are guaranteed to be non-singular and are particularly well-suited for iterative solution methods like Jacobi and Gauss-Seidel, which are guaranteed to converge for diagonally dominant systems.

66. Dimension

The number of vectors in any basis of a vector space, representing the minimum number of independent directions needed to describe every vector in the space. The dimension of Rⁿ is n. The dimension of the zero vector space is 0.

67. Dimension Theorem

See Rank-Nullity Theorem. The theorem stating that for an m × n matrix A, the rank plus the nullity equals n, the number of columns.

68. Direct Sum

A way of combining two subspaces W₁ and W₂ of a vector space V such that every vector in V can be written uniquely as a sum of a vector from W₁ and a vector from W₂, and the two subspaces intersect only at the zero vector. Written V = W₁ ⊕ W₂.

69. Distributive Property

Matrix multiplication distributes over matrix addition: A(B + C) = AB + AC and (A + B)C = AC + BC, provided the dimensions are compatible. This property holds for matrices even though multiplication is not generally commutative.

70. Dot Product

The sum of the products of corresponding components of two vectors of the same length. For vectors u and v in Rⁿ, u · v = u₁v₁ + u₂v₂ + … + uₙvₙ. The dot product is also written as uᵀv using matrix notation. It is zero if and only if the vectors are orthogonal.

71. Echelon Form

A matrix is in row echelon form if all zero rows are at the bottom, each leading non-zero entry (pivot) in a non-zero row is strictly to the right of the pivot in the row above, and all entries below a pivot in the same column are zero. It is the standard intermediate form in Gaussian elimination.

72. Eigendecomposition

The factorization of a diagonalizable matrix A into the form A = PDP⁻¹, where P contains the eigenvectors as columns and D is a diagonal matrix of eigenvalues. Eigendecomposition is a special case of the more general singular value decomposition.

73. Eigenspace

The set of all eigenvectors corresponding to a given eigenvalue λ, together with the zero vector. It is a subspace of Rⁿ equal to the null space of (A − λI). The dimension of an eigenspace is the geometric multiplicity of the corresponding eigenvalue.

74. Eigenvalue

A scalar λ such that there exists a non-zero vector x satisfying Ax = λx for a square matrix A. Eigenvalues are the roots of the characteristic polynomial det(λI − A) = 0. They represent scaling factors along specific directions defined by the corresponding eigenvectors.

75. Eigenvector

A non-zero vector x that satisfies Ax = λx for some scalar eigenvalue λ. An eigenvector is only scaled (not rotated) when multiplied by the matrix A. Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

76. Elementary Matrix

A matrix obtained from the identity matrix by performing exactly one elementary row operation. Multiplying a matrix A on the left by an elementary matrix performs the corresponding row operation on A. Every invertible matrix can be expressed as a product of elementary matrices.

77. Elementary Row Operations

The three basic operations that can be performed on the rows of a matrix without changing the solution set of the corresponding linear system. They are: swapping two rows, multiplying a row by a non-zero scalar, and adding a scalar multiple of one row to another row.

78. Elimination Method

A general term for methods that solve a linear system by systematically eliminating variables. Gaussian elimination and Gauss-Jordan elimination are the standard forms. The process uses elementary row operations to transform the augmented matrix into a simpler form.

79. Entry

A single element of a matrix, identified by its row and column indices. The entry in row i and column j of matrix A is written aᵢⱼ. The entry is also called a component or element of the matrix.

80. Equal Matrices

Two matrices A and B are equal if and only if they have the same dimensions and every corresponding pair of entries is equal: aᵢⱼ = bᵢⱼ for all i and j. Matrix equality requires both size and element-wise agreement.

81. Equivalent Row Operations

Elementary row operations that can be reversed and thus do not change the solution set of the associated linear system. Swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another are all reversible and equivalent in this sense.

82. Equivalent Systems

Two systems of linear equations that have exactly the same solution set. Applying elementary row operations to the augmented matrix of a system produces an equivalent system. Gaussian elimination produces a sequence of equivalent systems leading to a solved form.

83. Even Permutation

A permutation of a set of indices that can be expressed as a product of an even number of transpositions (swaps of two elements). The sign of an even permutation is +1. Determinants are defined using signed sums over permutations, alternating between even (+1) and odd (−1) contributions.

84. Exact Solution

A solution to a linear system expressed precisely without approximation, as opposed to a numerical solution obtained by iterative methods. Gaussian elimination and Cramer’s rule produce exact solutions when applied to systems with exact rational or symbolic coefficients.

85. Expansion by Cofactors

See Cofactor Expansion. The method of computing a determinant by expanding along a chosen row or column using the cofactors of the elements in that row or column.

86. Field

The algebraic structure formed by a set of scalars together with addition and multiplication operations satisfying specific axioms including commutativity, associativity, distributivity, and the existence of identity and inverse elements. Vector spaces are defined over fields, most commonly the real numbers or complex numbers.

87. Forward Elimination

The first phase of Gaussian elimination in which elementary row operations are applied to create zeros below each pivot, transforming the augmented matrix into upper triangular or row echelon form. Forward elimination proceeds from the first column to the last.

88. Free Variable

A variable in a linear system that is not associated with a pivot column in the row echelon form of the coefficient matrix. Free variables can take any value, and each free variable generates a direction in the solution space. The number of free variables equals the nullity of the coefficient matrix.

89. Full Rank Matrix

A matrix whose rank equals the smaller of its number of rows and columns. A square matrix of full rank is invertible. An m × n matrix with m < n is full rank if its rank equals m (all rows independent). An m × n matrix with n < m is full rank if its rank equals n (all columns independent).

90. Fundamental Matrix

A square matrix whose columns are linearly independent solutions to a system of linear differential equations. It plays a key role in expressing the general solution of a linear differential system and is used to compute the matrix exponential.

91. Fundamental Subspaces

The four subspaces naturally associated with any matrix A: the column space, the null space, the row space, and the left null space. These four subspaces are related through the rank-nullity theorem and the orthogonality of their pairings.

92. Gauss-Jordan Elimination

An extension of Gaussian elimination that continues row reduction beyond echelon form to reach reduced row echelon form. It eliminates entries both below and above each pivot, producing a matrix that directly reveals the solution without back substitution.

93. Gauss-Seidel Method

An iterative method for solving a square linear system in which each unknown is updated using the most recently computed values of all other unknowns. It often converges faster than the Jacobi method and is commonly used for large sparse systems in engineering simulation.

94. Gaussian Elimination

The standard algorithm for solving systems of linear equations by applying elementary row operations to the augmented matrix to reduce it to row echelon form, followed by back substitution to find the solution. It is the most widely used direct method for linear systems.

95. General Solution

The complete set of all solutions to a linear system, expressed as the sum of a particular solution to the non-homogeneous system and the general solution to the associated homogeneous system. The general solution describes every possible solution using free variables as parameters.

96. Generalized Eigenvector

A vector x satisfying (A − λI)ᵏx = 0 for some positive integer k, but not satisfying (A − λI)^(k−1)x = 0. Generalized eigenvectors arise for defective matrices and are used to construct the Jordan normal form and the chains needed for the general solution of linear differential systems.

97. Geometric Multiplicity

The dimension of the eigenspace corresponding to a given eigenvalue, equal to the number of linearly independent eigenvectors associated with that eigenvalue. The geometric multiplicity is always less than or equal to the algebraic multiplicity.

98. Gram-Schmidt Process

An algorithm that transforms a set of linearly independent vectors into an orthonormal set spanning the same subspace. At each step, a new vector is made orthogonal to all previous vectors by subtracting projections, then normalized to unit length. It is the basis of QR decomposition.

99. Hermitian Matrix

A complex square matrix that equals its own conjugate transpose: A = A*. The diagonal entries of a Hermitian matrix are always real. Hermitian matrices are the complex generalization of symmetric matrices and have real eigenvalues.

100. Homogeneous Coordinates

A system of coordinates used in projective geometry where a point in n-dimensional space is represented by an (n + 1)-dimensional vector. Homogeneous coordinates allow translations (which are not linear transformations) to be represented as matrix multiplications, making them essential in computer graphics and robotics.

251. Zero Vector

The vector with all components equal to zero, denoted 0 or 0. It is the additive identity in any vector space: v + 0 = v for all vectors v. The zero vector is the unique vector that belongs to every subspace of a given vector space.

Conclusion

Matrices and linear algebra is both a computational and a conceptual subject. On the board exam, the computational skills tested most often are determinant evaluation (2 × 2 and 3 × 3), matrix operations and the inverse formula for 2 × 2 matrices, solving systems using Gaussian elimination or Cramer’s rule, and finding eigenvalues and eigenvectors of small matrices. These are the high-frequency items that show up repeatedly across different years and different engineering disciplines. Master these computations first, because they are the foundation on which everything else is built.

After the computations, focus on understanding the structural concepts: rank, nullity, linear independence, span, basis, and dimension. The Rank-Nullity Theorem connects these ideas into a single powerful statement about how a matrix transforms space. Understanding this theorem conceptually allows you to answer questions about the number of solutions to a system, the dimensions of null spaces and column spaces, and the conditions for invertibility without performing full computations. These conceptual questions appear in the exam both directly and embedded in applied problems, and they reward reviewees who have taken the time to understand the definitions in this glossary.

Finally, connect the linear algebra you know to the engineering contexts where it appears. Eigenvalues and eigenvectors describe the natural frequencies and modes of mechanical and structural systems. Matrix equations describe circuit networks and power flow problems. Least-squares solutions underlie data fitting and system identification. The stronger your command of the vocabulary and theory in this glossary, the faster you can recognize the structure of a problem and apply the right tool. That speed and recognition, built through solid review, is what produces passing scores on the Mathematics engineering board exam.

For practice problems on all these topics, head over to our Advanced Mathematics Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.