Source: Friedberg, Insel & Spence, Linear Algebra 4th Ed., Ch. 1
Tags: vector spaces, linear algebra, vector addition, scalar multiplication, parallelogram law, n-tuple, matrix, polynomial, field, VS axioms, UIUC abstract algebra
Difficulty: Foundational Prerequisites: Basic familiarity with coordinate geometry and real numbers. Comfort with algebraic manipulation.
This is the opening chapter of an abstract linear algebra course, and everything that follows in the subject rests on what a vector space is. You are moving from the geometric idea of a "vector as an arrow" to a general algebraic structure defined by axioms. If you missed the first few weeks, start here: you need to understand the eight axioms (VS 1–VS 8), and you need to recognise that "vector" no longer means just an arrow in space. Polynomials, matrices, functions, and sequences can all be vectors if the right operations are defined.
A vector space is a set V over a field F equipped with addition and scalar multiplication satisfying eight axioms (commutativity, associativity, identity, inverses, distributivity, etc.). The key examples are F^n (n-tuples), M_{m×n}(F) (matrices), P(F) (polynomials), and F(S,F) (functions from a set to a field). The geometric arrow-based picture of vectors is just one special case of this broader structure.
Vector (geometric sense)
An entity with both magnitude and direction, represented by an arrow. Two vectors are equal if they share the same magnitude and direction, regardless of position.
In simple terms, think of a vector as an arrow you can slide around freely without changing what it "is."
Parallelogram law
The rule for adding two vectors: the sum of vectors x and y acting at the same point P is the diagonal of the parallelogram with x and y as adjacent sides.
Think of it as: place two arrows tail-to-tail, complete the parallelogram, and the diagonal from the shared tail is the sum.
Scalar multiplication
Multiplying a vector by a real number (scalar) t. The result tx has length |t| times the original length, points in the same direction if t ≥ 0, and in the opposite direction if t < 0.
In simple terms, stretching or shrinking an arrow, and flipping it if the scalar is negative.
Parallel vectors
Two nonzero vectors x and y are parallel if y = tx for some nonzero real number t.
Think of it as: one vector is just a scaled (and possibly reversed) version of the other.
Vector space (linear space)
A set V over a field F with two operations, addition and scalar multiplication, satisfying all eight axioms VS 1–VS 8 (listed below).
In simple terms, a collection of objects where you can add any two together and multiply any one by a scalar, and all the "sensible" algebraic rules still hold.
Field
A set of scalars (like the real numbers R or complex numbers C) where addition, subtraction, multiplication, and division (by nonzero elements) are all defined and well-behaved. The field provides the scalars for a vector space.
Think of it as: the "number system" that your scalars live in.
n-tuple
An ordered list (a₁, a₂, ..., aₙ) of n entries from a field F. The entries are also called components.
In simple terms, a list of n numbers in a fixed order.
F^n
The set of all n-tuples with entries from a field F. It is a vector space under coordinatewise addition and scalar multiplication.
Think of it as: the generalisation of R² and R³ to n dimensions.
Matrix
A rectangular array of entries from a field F arranged in m rows and n columns, written as an m × n matrix.
In simple terms, a grid of numbers.
M_{m×n}(F)
The vector space of all m × n matrices with entries from F, using entrywise addition and scalar multiplication.
Zero matrix (O)
The m × n matrix in which every entry is zero. This is the additive identity in M_{m×n}(F).
Square matrix
A matrix where the number of rows equals the number of columns.
Diagonal entries
The entries A_{ij} of a matrix where i = j (i.e. row index equals column index).
Polynomial
An expression f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ with coefficients from a field F.
Degree of a polynomial
The largest exponent of x with a nonzero coefficient. The zero polynomial is assigned degree −1 by convention.
P(F)
The vector space of all polynomials with coefficients from F, under standard polynomial addition and scalar multiplication.
P_n(F)
The subset of P(F) consisting of polynomials of degree at most n. (This is itself a vector space, covered in Section 1.3.)
F(S, F)
The vector space of all functions from a nonempty set S to a field F, with pointwise addition and scalar multiplication: (f + g)(s) = f(s) + g(s) and (cf)(s) = c·f(s).
Zero vector (0)
The unique element in V satisfying x + 0 = x for every x in V. Uniqueness is proved via the cancellation law.
Additive inverse (−x)
The unique vector y such that x + y = 0. Equals (−1)x by Theorem 1.2.
For a set V over a field F with addition (+) and scalar multiplication (·):
VS 1 (Commutativity of addition): x + y = y + x for all x, y in V
VS 2 (Associativity of addition): (x + y) + z = x + (y + z) for all x, y, z in V
VS 3 (Additive identity): There exists 0 in V such that x + 0 = x for all x in V
VS 4 (Additive inverses): For each x in V, there exists y in V with x + y = 0
VS 5 (Multiplicative identity): 1x = x for all x in V
VS 6 (Associativity of scalar multiplication): (ab)x = a(bx) for all a, b in F and x in V
VS 7 (Distributivity over vector addition): a(x + y) = ax + ay
VS 8 (Distributivity over scalar addition): (a + b)x = ax + bx
Any structure satisfying all eight is a vector space. Any structure failing even one is not.
F^n: n-tuples with coordinatewise operations. R³ is the familiar 3D space.
Addition: (a₁,...,aₙ) + (b₁,...,bₙ) = (a₁+b₁,...,aₙ+bₙ)
Scalar multiplication: c(a₁,...,aₙ) = (ca₁,...,caₙ)
M_{m×n}(F): Matrices with entrywise operations.
(A + B){ij} = A{ij} + B_{ij} and (cA){ij} = cA{ij}
P(F): Polynomials with standard addition and scalar multiplication of coefficients.
F(S, F): Function spaces with pointwise operations. This generalises to continuous functions, differentiable functions, etc.
Sequences: {aₙ} + {bₙ} = {aₙ + bₙ} and t{aₙ} = {taₙ}. The set of sequences with finitely many nonzero terms forms a vector space.
Example 6 from the text: S = R² with addition (a₁,a₂) + (b₁,b₂) = (a₁+b₁, a₂−b₂) and standard scalar multiplication. Fails VS 1, VS 2, and VS 8. The modified addition breaks commutativity and associativity.
Example 7 from the text: S = R² with addition (a₁,a₂) + (b₁,b₂) = (a₁+b₁, 0) and scalar multiplication c(a₁,a₂) = (ca₁, 0). Fails VS 3 (no proper zero vector), hence VS 4 fails too, and VS 5 also fails.
Theorem 1.1 (Cancellation law):
If x + z = y + z, then x = y.
Corollary 1: The zero vector is unique.
Corollary 2: The additive inverse of each vector is unique.
Theorem 1.2 (Properties of scalar multiplication):
0x = 0 for every vector x (the scalar zero times anything is the zero vector)
(−a)x = −(ax) = a(−x) for all scalars a and vectors x
a·0 = 0 for every scalar a (any scalar times the zero vector is the zero vector)
Vectors in a plane add by the parallelogram law, described algebraically as coordinatewise addition.
The equation of a line through points A and B in space is x = u + t(v − u), where u and v are position vectors of A and B.
The equation of a plane through three noncollinear points A, B, C in space is x = A + su + tv, where u and v are vectors from A to B and from A to C.
Line through two points A, B:
x = u + t(v − u), where t is a real parameter
Plane through three noncollinear points A, B, C:
x = A + su + tv, where s and t are real parameters
Coordinatewise addition in R^n:
(a₁,...,aₙ) + (b₁,...,bₙ) = (a₁+b₁,...,aₙ+bₙ)
Coordinatewise scalar multiplication in R^n:
c(a₁,...,aₙ) = (ca₁,...,caₙ)
Vectors model forces, velocities, and accelerations in physics, where the parallelogram law captures how two forces combine. The abstract framework of vector spaces goes far beyond physics: signal processing treats audio signals as vectors in a function space, and computer graphics relies on matrix vector spaces for transformations.
Students often think "vector" only means an arrow in 2D or 3D space. In abstract algebra, a vector is any element of a vector space, which could be a polynomial, a matrix, or a function.
It is easy to confuse the scalar zero (0 in F) with the zero vector (0 in V). They are different objects. The statement 0x = 0 has the scalar 0 on the left and the vector 0 on the right.
Students sometimes assume that if two sets have the same underlying elements, any operations you define on them will yield a vector space. The operations must satisfy all eight axioms; changing even one operation can break the structure entirely.
The field matters. R² over R and C¹ over C are different vector spaces, even though they look similar.
⚠️ You will be asked to verify whether a given set with given operations forms a vector space. This means checking all eight axioms (or finding one that fails).
⚠️ The cancellation law and uniqueness of the zero vector and additive inverses are standard proof questions.
⚠️ Know the key examples (F^n, M_{m×n}(F), P(F), F(S,F)) and be able to state their operations precisely.
⚠️ Theorem 1.2 (0x = 0, (−1)x = −x, a·0 = 0) appears in proofs throughout the course. Know how to derive these from the axioms.
True or false: every vector space contains a zero vector.
True or false: a vector space may have more than one zero vector.
Fill in the blank: in any vector space, (−1)x = ______.
True or false: an m × n matrix has m columns and n rows.
True or false: if f and g are polynomials of degree n, then f + g is always a polynomial of degree n.
Answers: 1. True. 2. False (Corollary 1 of Thm 1.1). 3. −x. 4. False (m rows and n columns). 5. False (leading terms could cancel).
Q: State the eight vector space axioms.
A: VS 1 (commutativity of addition), VS 2 (associativity of addition), VS 3 (existence of zero vector), VS 4 (existence of additive inverses), VS 5 (1x = x), VS 6 (associativity of scalar multiplication), VS 7 (distributivity of scalar mult. over vector addition), VS 8 (distributivity of scalar mult. over scalar addition).
Q: Prove that the zero vector in a vector space is unique.
A: Suppose 0 and 0' are both zero vectors. Then 0 = 0 + 0' = 0' + 0 = 0'. The first equality uses VS 3 with 0' as the zero vector, and the last uses VS 3 with 0 as the zero vector. So 0 = 0'.
Q: Show that 0x = 0 for any vector x in a vector space V.
A: 0x + 0x = (0 + 0)x = 0x = 0x + 0 by VS 8, VS 3. By the cancellation law (Thm 1.1), 0x = 0.
Q: Let S = {(a₁, a₂) : a₁, a₂ ∈ R} with (a₁,a₂) + (b₁,b₂) = (a₁+b₁, 0) and c(a₁,a₂) = (ca₁, 0). Explain why S is not a vector space.
A: VS 5 fails: 1·(a₁, a₂) = (a₁, 0) ≠ (a₁, a₂) whenever a₂ ≠ 0. Also, VS 3 fails because there is no element (z₁, z₂) in S with (a₁, a₂) + (z₁, z₂) = (a₁, a₂) for all (a₁, a₂), since the second component always collapses to 0.
Q: Write the equation of the plane containing the points A = (1, 0, 2), B = (−3, −2, 4), and C = (1, 8, −5).
A: u = B − A = (−4, −2, 2), v = C − A = (0, 8, −7). The equation is x = (1, 0, 2) + s(−4, −2, 2) + t(0, 8, −7).
This material connects directly to subspaces (Section 1.3), which are subsets of a vector space that are themselves vector spaces under the inherited operations. It also sets up linear combinations and span (Section 1.4), where you ask which vectors can be built from a given set. The abstract definition of a vector space reappears in linear transformations (Chapter 2), where you study functions between vector spaces that respect the operations.
vector space definition, linear space, field, scalar, vector addition, scalar multiplication, VS axioms, n-tuple, Fn, matrix space, polynomial space, function space, zero vector, additive inverse, cancellation law, parallelogram law, UIUC abstract linear algebra, Friedberg Insel Spence Chapter 1