Mathematical Notation and Language
Subtopic: Notations and Language
Topic: Linear algebra
The Language of Linear Algebra
Before diving into the theorems and computations of linear algebra, you must first learn its language. Mathematics uses precise notation to communicate ideas unambiguously. A single symbol like A can represent an entire matrix of numbers. Understanding this notation is not just about memorizing symbols — it is about learning to read and write mathematics fluently.
Mathematical notation serves three purposes. First, it compresses information: writing Ax = b is far more efficient than describing a system of n equations in n unknowns. Second, it reveals structure: the notation makes patterns visible that would be hidden in lengthy descriptions. Third, it enables manipulation: once you express relationships symbolically, you can apply algebraic rules to transform and solve problems.
Vector Notation
A vector is an ordered list of numbers. In linear algebra, vectors are fundamental objects that can represent points in space, directions, states of a system, or any quantity that has multiple components.
The standard convention writes vectors as column vectors — vertical arrays of numbers stacked on top of each other:
Here v₁, v₂, ..., vₙ are the components or entries of the vector. The subscript indicates position: v₁ is the first component, v₂ is the second, and so on. A vector with n components lives in n-dimensional space, written ℝⁿ (pronounced "R-n").
Sometimes, especially in inline text, we write vectors horizontally using parentheses or angle brackets:
This horizontal notation is understood to still represent a column vector unless explicitly stated otherwise.
Bold vs Arrow Notation
Two conventions exist for distinguishing vectors from scalars in written mathematics.
The bold convention uses boldface letters for vectors and regular (italic) letters for scalars:
This convention dominates in linear algebra textbooks, physics, engineering, and computer science. When you see a bold letter, you know immediately that it represents a vector, not a single number.
The arrow convention places an arrow above the letter:
This notation is more common in introductory physics courses, particularly when discussing displacement, velocity, and force. The arrow visually suggests direction, which connects to the geometric interpretation of vectors.
Both notations convey the same mathematical meaning. The choice is largely stylistic, though bold is easier to write on chalkboards (by doubling vertical strokes) and more common in advanced mathematics. This text uses bold notation throughout.
When handwriting, since true boldface is impractical, mathematicians often write vectors with an underline or a wavy underline:
Matrix Notation
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are denoted by capital letters, typically in bold or regular italic:
A, B, M, P \quad \text{or} \quad \mathbf{A}, \mathbf{B}, \mathbf{M}, \mathbf{P}
An m × n matrix (read "m by n") has m rows and n columns. We write:
to indicate that A is a matrix with real entries having m rows and n columns. The general form of such a matrix is:
The entry in row i and column j is written aᵢⱼ (or a_ij). The first subscript always indicates the row; the second indicates the column. This convention is universal and must be memorized: row first, column second.
We can write a matrix compactly as:
This notation indicates that A is the matrix whose (i,j) entry is aᵢⱼ.
Columns and Rows of a Matrix
A matrix can be viewed as a collection of column vectors placed side by side, or as a stack of row vectors.
The column notation writes A in terms of its columns a₁, a₂, ..., aₙ:
Each aⱼ is a column vector in ℝᵐ. When we write A this way, we emphasize that a matrix is built from its columns.
The row notation writes A in terms of its rows:
Each rᵢ is a row vector with n components. Row vectors are 1 × n matrices.
These two viewpoints — column picture and row picture — give complementary insights. When multiplying Ax, the column picture sees the result as a linear combination of the columns of A. The row picture sees it as dot products of the rows of A with x.
The Transpose
The transpose of a matrix A, denoted by a superscript T, is obtained by interchanging rows and columns. What was in row i, column j is now in row j, column i:
If A is an m by n matrix, then its transpose is an n by m matrix. Transposing flips the matrix across its main diagonal.
For example:
The transpose operation has several fundamental properties. For any matrices A and B (when dimensions are compatible) and scalar c:
The last property is crucial and often surprising: when transposing a product, the order reverses. This is not a typo — it must be B transpose times A transpose, not the other way around.
For vectors, the transpose converts column vectors to row vectors and vice versa. If v is a column vector in the space of real n-tuples:
This notation is essential for expressing the dot product (inner product) of two vectors. If u and v are column vectors in the same space:
The product of the row vector u transpose with the column vector v produces a scalar.
Symmetric and Skew-Symmetric Matrices
A square matrix A is symmetric if it equals its own transpose:
A = A^T \quad \text{(symmetric)}
Symmetric matrices have the property that entries are mirrored across the main diagonal. If you know the upper triangle, you know the whole matrix.
A square matrix A is skew-symmetric (or antisymmetric) if:
A = -A^T \quad \text{(skew-symmetric)}
For a skew-symmetric matrix, the diagonal entries must all be zero (since each diagonal entry equals its own negative), and off-diagonal entries are negatives of their mirror images.
Set Membership: The Symbol ∈
The symbol ∈ means "is an element of" or "belongs to." It asserts membership in a set. When we write:
we are stating that v is a vector in n-dimensional real space — that is, v is a column of n real numbers.
The symbol ℝ denotes the set of all real numbers. The superscript n indicates dimension: ℝ² is the plane, ℝ³ is three-dimensional space, and ℝⁿ is n-dimensional space.
For matrices, we write:
to indicate that A is a matrix with m rows and n columns, all entries being real numbers.
The negation of set membership is written with the symbol ∉, meaning "is not an element of."
Other important number sets in linear algebra include:
\mathbb{C} \quad \text{(complex numbers)}
\mathbb{Q} \quad \text{(rational numbers)}
\mathbb{Z} \quad \text{(integers)}
\mathbb{N} \quad \text{(natural numbers)}
The double-struck letters are called blackboard bold. They originated from the practice of writing bold letters on chalkboards by doubling vertical strokes.
Universal Quantifier: ∀
The symbol ∀ is the universal quantifier, read as "for all" or "for every." It makes statements about every element of a set.
For example, the definition of a linear transformation T can be written:
This reads: "For all vectors u and v in V, and for all scalars c in the real numbers, the transformation T preserves addition and scalar multiplication."
The universal quantifier appears frequently in definitions and theorems. When a property holds "for all" elements, it is a universal statement.
Existential Quantifier: ∃
The symbol ∃ is the existential quantifier, read as "there exists" or "for some." It asserts that at least one element with a given property exists.
For example, the statement that a system Ax = b has a solution can be written:
This reads: "There exists a vector x in n-dimensional real space such that Ax equals b."
The variant ∃! (exists unique) asserts that exactly one such element exists:
This reads: "There exists a unique vector x such that Ax = b."
A key result in linear algebra: when A is an invertible n by n matrix, for every b there exists a unique solution x to Ax = b.
Combining Quantifiers
Quantifiers can be combined, and the order matters. Consider the difference between:
This says: "For every b, there exists some x such that Ax = b." The x can depend on which b we choose.
This says: "There exists a single x that works for every b." This is generally false — one x cannot simultaneously satisfy Ax = b for all possible b.
Swapping the order of quantifiers changes the meaning dramatically. Always read quantifier statements carefully.
Greek Letters in Linear Algebra
Greek letters pervade mathematical notation. In linear algebra, certain Greek letters have conventional meanings. Learning these conventions helps you read mathematics fluently.
Scalars
Scalars (individual numbers used to scale vectors) are often denoted by Greek letters:
The eigenvalue lambda is particularly important. When we write the eigenvalue equation:
the scalar lambda is the eigenvalue and v is the eigenvector. The choice of lambda for eigenvalues is universal across mathematics and physics.
Matrices and Transformations
Certain Greek capital letters denote matrices with special roles:
When a matrix A is diagonalizable, we often write its eigenvalue decomposition as:
where Lambda is the diagonal matrix containing the eigenvalues and P is the matrix of eigenvectors.
Similarly, the singular value decomposition uses Sigma for singular values:
Angles and Trigonometry
When angles arise in linear algebra (particularly in geometry, rotations, and orthogonality), these Greek letters are standard:
A rotation matrix in 2D by angle theta takes the form:
Summation and Product Notation
The capital Greek letters Sigma and Pi have special roles as mathematical operators:
Summation appears constantly in linear algebra. Matrix-vector multiplication, dot products, and traces all involve sums. The dot product of two vectors is:
The entry in row i, column j of a matrix product AB is:
The trace of a matrix (sum of diagonal entries) is:
Indices and Subscripts
In linear algebra, we constantly work with indexed quantities. Understanding indexing conventions is crucial.
For vectors, a single subscript indicates the component:
v_i \quad \text{or} \quad (\mathbf{v})_i \quad - \quad \text{the } i\text{-th component of vector } \mathbf{v}
For matrices, double subscripts indicate row and column:
a_ij \quad \text{or} \quad (A)_ij \quad - \quad \text{the entry in row } i \text{, column } j \text{ of matrix } A
When we have multiple vectors or matrices, we use superscripts or subscripts to distinguish them:
When both indexing conventions appear together, superscripts often denote the vector number while subscripts denote the component:
v_i^(j) \quad - \quad \text{component } i \text{ of vector } j
Norms and Absolute Values
The norm of a vector measures its length or magnitude. Several notations appear:
The double-bar notation is more common in linear algebra to distinguish from absolute value. The Euclidean norm (or 2-norm) is:
Other important norms include:
A unit vector is a vector with norm 1. To normalize a nonzero vector (convert it to a unit vector pointing in the same direction):
The hat notation (circumflex) commonly denotes unit vectors.
Special Matrices
Certain matrices have standard notation:
I \quad \text{or} \quad I_n \quad - \quad \text{identity matrix (n by n)}
O \quad \text{or} \quad \mathbf{0} \quad - \quad \text{zero matrix}
A^-1 \quad - \quad \text{inverse of A}
A^T \quad - \quad \text{transpose of A}
A^* \quad \text{or} \quad A^H \quad - \quad \text{conjugate transpose (Hermitian adjoint)}
The identity matrix has ones on the diagonal and zeros elsewhere:
The identity matrix is the multiplicative identity: AI = IA = A for any compatible matrix A.
Other Important Symbols
Several other symbols appear regularly in linear algebra:
\det(A) \quad \text{or} \quad |A| \quad - \quad \text{determinant of A}
\dim(V) \quad - \quad \text{dimension of vector space V}
\ker(A) \quad \text{or} \quad \text{null}(A) \quad - \quad \text{kernel (null space) of A}
\text{im}(A) \quad \text{or} \quad \text{range}(A) \quad \text{or} \quad \text{col}(A) \quad - \quad \text{image (column space) of A}
Reading Mathematical Statements
Mathematical statements combine symbols into precise claims. Consider this statement about invertible matrices:
This reads: For all n by n real matrices A, the determinant of A is nonzero if and only if there exists an n by n matrix (called A inverse) such that the product of A and its inverse in either order equals the identity matrix.
The double arrow symbol means "if and only if" — both directions hold. If the determinant is nonzero, the inverse exists. And if the inverse exists, the determinant must be nonzero.
Building Fluency
Learning mathematical notation is like learning a language. At first, you translate symbol by symbol. With practice, you read entire expressions at once, grasping their meaning immediately.
The key to building fluency is active engagement. When you encounter a new symbol, write it out, use it in examples, and say it aloud. The notation is not arbitrary — each symbol was chosen because it helps mathematicians communicate efficiently.
As you work through linear algebra, you will encounter these symbols repeatedly. Each encounter strengthens your fluency. Eventually, expressions like the eigenvalue equation or the singular value decomposition will become as readable as ordinary sentences.