Theory

Vector spaces, bases and dimension

The common structure of vectors, matrices and polynomials, subspaces described by generators or by equations, bases, dimension and Grassmann's formula for sum and intersection.

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Vector spaces
Vector subspaces

Vector spaces

Real vector space

A real vector space is a set $V$ equipped with two operations, addition $V \times V \to V$ , $(u, v) \mapsto u + v$ , and scalar multiplication $R \times V \to V$ , $(λ, v) \mapsto λ v$ , such that the following properties hold for every $u, v, w \in V$ and every $λ, μ \in R$ .

Addition is associative and commutative: $(u + v) + w = u + (v + w)$ and $u + v = v + u$ .
There exists a zero vector $0 \in V$ with $v + 0 = v$ for every $v$ .
Every $v$ has an opposite $- v$ with $v + (- v) = 0$ .
Distributivity: $λ (u + v) = λ u + λ v$ and $(λ + μ) v = λ v + μv$ .
Compatibility: $λ (μv) = (λ μ) v$ and $1 \cdot v = v$ .

The elements of $V$ are called vectors, and the real numbers are called scalars. The definition collects only the computational rules needed to speak of linear combinations, bases and dimension; therefore every result proved from them holds simultaneously for $n$ -tuples, for matrices and for polynomials.

The vector spaces of the exam

$R^{n}$ , with componentwise addition and scalar multiplication: $(1, 2, 0) + 3 (0, 1, 1) = (1, 5, 3)$ .
$M_{m, n} (R)$ , the $m \times n$ matrices with entrywise addition and scalar multiplication; the zero vector is the zero matrix.
$R_{n} [x]$ , the polynomials with real coefficients of degree less than or equal to $n$ , including the zero polynomial, with the usual operations on polynomials.

For $R_{n} [x]$ the condition on the degree is written as an inequality because the polynomials of degree exactly $2$ do not form a vector space: the sum $(x^{2} + x) + (- x^{2}) = x$ has degree $1$ .

Computational rules

The axioms imply the rules $0 \cdot v = 0$ , $λ \cdot 0 = 0$ and $(- 1) v = - v$ . For the first one we write $0 \cdot v = (0 + 0) v = 0 \cdot v + 0 \cdot v$ ; adding the opposite of $0 \cdot v$ to both sides gives $0 = 0 \cdot v$ . Moreover, the zero-product law holds: if $λ v = 0$ , then $λ = 0$ or $v = 0$ . These rules are used in all subsequent computations without being cited.

Vector subspaces

1. Vector subspace

A subset $W \subseteq V$ is a vector subspace of $V$ if it is itself a vector space with the operations of $V$ . Every space $V$ contains the trivial subspaces ${0}$ and $V$ . In $R^{2}$ the other subspaces are the lines through the origin, in $R^{3}$ the lines and the planes through the origin, as the computation of the dimension will show.

2. Subspace criterion

Let $V$ be a vector space and let $W \subseteq V$ . Then $W$ is a subspace of $V$ if and only if the following three conditions are satisfied together:

⎩ ⎨ ⎧ 0 \in W u + v \in W for every u, v \in W λ u \in W for every u \in W, λ \in R

The criterion reduces the verification of the axioms to three checks. To rule out that $W$ is a subspace, a counterexample to one of the three conditions is sufficient, and the quickest one to check is $0 \in W$ .

Proof

If $W$ is a subspace, the sum and the scalar multiple of elements of $W$ lie in $W$ by definition, and $0 \in W$ since for every $u \in W$ we have $0 = 0 \cdot u \in W$ . Conversely, if the three conditions are satisfied, the two operations of $V$ remain operations on $W$ . The associative, commutative, distributive and compatibility properties hold in $W$ because they hold in $V$ ; the zero vector lies in $W$ by the first condition, and the opposite of $u \in W$ is $- u = (- 1) u$ , which lies in $W$ by the third. Therefore $W$ is a vector space.

A plane through the origin

$W = {(x, y, z) \in R^{3} : x + y - 2 z = 0}$ . The zero vector satisfies the equation. If $u = (u_{1}, u_{2}, u_{3})$ and $v = (v_{1}, v_{2}, v_{3})$ lie in $W$ , then

(u_{1} + v_{1}) + (u_{2} + v_{2}) - 2 (u_{3} + v_{3}) = (u_{1} + u_{2} - 2 u_{3}) + (v_{1} + v_{2} - 2 v_{3}) = 0 + 0 = 0,

so $u + v \in W$ ; in the same way $λ u_{1} + λ u_{2} - 2 λ u_{3} = λ \cdot 0 = 0$ , so $λ u \in W$ . By the subspace criterion, $W$ is a subspace of $R^{3}$ .

Homogeneous and non-homogeneous equations

The argument of the example holds for every homogeneous linear system: the solution set of $A x = 0$ , with $A \in M_{m, n} (R)$ , is a subspace of $R^{n}$ , since $A 0 = 0$ , $A (u + v) = A u + A v$ and $A (λ u) = λ A u$ . The solution set of a system $A x = b$ with $b \neq = 0$ is not a subspace, because the zero vector does not satisfy it. In the plane the line $x - 2 y = 0$ is a subspace, while the parallel line $x - 2 y = - 2$ does not contain the origin.

y = \dfrac{x}{2}

y = \dfrac{x}{2} + 1

The line

x - 2 y = 0

passes through the origin and is a subspace of

R^{2}

; the parallel line

x - 2 y = - 2

(dashed) does not contain the zero vector

Warning

A subset can contain the zero vector and be closed under scalar multiplication without being a subspace. The union of the two axes of $R^{2}$ , that is ${(x, y) : x y = 0}$ , contains $(1, 0)$ and $(0, 1)$ , while their sum $(1, 1)$ does not satisfy $x y = 0$ . The three conditions of the criterion must all be verified.

The set

x y = 0

(in color) is the union of the two axes: it contains

u = (1, 0)

and

v = (0, 1)

, and it does not contain the sum

u + v = (1, 1)

The theory continues with 8 more sections (linear combinations and generators, linear dependence and independence, bases and coordinates…), plus all worked examples, the solutions and the PDF.

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