We now arrive at the central structure of linear algebra. A vector space is a set with two operations, addition and scalar multiplication, satisfying axioms modelled on the familiar properties of . Throughout, denotes a field.
Definition 4.5.1 (Vector space). Fix a field . A vector space over (or -vector space) is a set equipped with two operations, vector addition and scalar multiplication , , satisfying the following axioms:
- is an abelian group, whose identity is called the zero vector;
- for all ;
- for all and ;
- for all and ;
- for all and .
Elements of are called vectors; elements of are called scalars.
Axioms (b) and (c) express that acts as the identity and that scalar multiplication is associative; the two distributive laws (d) and (e) connect scalar multiplication with the two additions.
Proposition 4.5.2 (First consequences of the axioms). Let be a vector space over , and . Then:
- and ;
- ;
- if , then or .
Proof. (a) We have ; adding to both sides gives . Similarly , so .
(b) .
(c) If and , then . ∎
4.5.1 Fundamental Examples
Example 4.5.3. The standard vector space : For , the set is a vector space with componentwise operations: We also set . This is the prototypical -dimensional vector space.
Example 4.5.4. Infinite sequences : The set of all sequences in , is a vector space with componentwise operations. It contains the eventually zero sequences, those with for all sufficiently large , which form a vector space under the same operations.
Example 4.5.5. Function spaces: For any set , the set of all functions from to is a vector space with pointwise operations: and . When we recover , and when we recover .
Example 4.5.6. Polynomials and power series:
- is a -vector space: addition is polynomial addition, and scalar multiplication multiplies each coefficient by the scalar.
- is a -vector space in the same way.
- For each , the set of polynomials of degree at most is a -vector space.
Example 4.5.7. Spaces of functions : The space of all functions contains many interesting vector spaces, each closed under addition and scalar multiplication:
- the continuous functions ;
- the differentiable functions;
- the continuously differentiable functions ;
- the smooth (infinitely differentiable) functions ;
- the polynomial functions .
Each is contained in the one before it. A polynomial defines the polynomial function , and two different polynomials define different functions: their difference is a nonzero polynomial, which by Corollary 4.3.15 has only finitely many roots, while is infinite. So we may identify with the space of real polynomial functions. Over a finite field this identification fails: over , the polynomials and define the same function , because for every in the group of order (a corollary of Lagrange’s theorem).
4.5.2 Subspaces
Definition 4.5.8 (Subspace). A subspace of a vector space is a nonempty subset that is closed under addition and scalar multiplication:
- ;
- .
A subspace is a vector space under the operations inherited from .
Proposition 4.5.9 (Subspace criterion). A nonempty subset is a subspace if and only if for all and .
Proof. () If is a subspace, then and by closure under scalar multiplication, and by closure under addition.
() Taking gives closure under addition. Taking gives closure under scalar multiplication. ∎
Note. A subspace always contains : if , then by Proposition 4.5.2. So in the definition we may replace “nonempty” by “contains ”. Some condition of this kind is needed, since the empty set is closed under both operations but is not a vector space. With it, a subspace is precisely a subset that contains and is closed under linear combinations, and it contains along with each .
Example 4.5.10.
- and itself are always subspaces of .
- In , every line through the origin and every plane through the origin is a subspace. Once we have the notion of dimension (Section 4.7), it will follow that, together with and , these are the only subspaces of .
- The eventually zero sequences form a subspace of , and is a subspace of .
- In , the polynomials of degree at most form a subspace. The polynomials of degree exactly do not: the zero polynomial is missing, and for the set is not closed under addition, since .
Proposition 4.5.11 (Intersections of subspaces). The intersection of any nonempty family of subspaces of is a subspace of .
Proof. Let . Each contains , so . If and , then for every , so . ∎
4.5.3 Linear Maps and Isomorphisms
To compare vector spaces we need maps that respect their structure. Chapter 5 studies these maps in detail; here we introduce them, since we need the notion of isomorphism right away.
Definition 4.5.12 (Linear map). Let and be vector spaces over . A linear map (or -linear map, or homomorphism of vector spaces) is a function such that
- for all ;
- for all and .
A bijective linear map is an isomorphism. We say and are isomorphic, written , if there is an isomorphism .
By induction on , a linear map preserves all linear combinations: In particular .
Proposition 4.5.13 (Composites and inverses). The composite of linear maps is linear, and the inverse of an isomorphism is linear. Consequently, is an equivalence relation.
Proof. If and are linear, then . If is a linear bijection, and , then , so . Hence is reflexive (via ), symmetric (via inverses) and transitive (via composites). ∎
Example 4.5.14. : The map is bijective, since a power series is nothing but its sequence of coefficients, and it is linear, since both spaces add and scale coefficientwise. It is therefore an isomorphism of vector spaces. It restricts to an isomorphism from onto the subspace of eventually zero sequences. The isomorphism ignores the multiplication of power series, which is not coefficientwise: as vector spaces the two are the same, as rings they are not.
4.5.4 Sums and Direct Sums of Subspaces
When can we decompose a vector space into independent pieces? The notion of direct sum captures exactly this.
Definition 4.5.15 (Sum of subspaces). Given subspaces , their sum is More generally, the sum of subspaces is .
Proposition 4.5.16 (Sum is a subspace). For subspaces , the sum is a subspace of , and it is the smallest subspace of containing both and .
Proof. The zero vector lies in . Let and be elements of , and let . Then since and . Similarly, . It contains (take ) and (take ), and any subspace containing and contains all sums . ∎
The same argument shows that is the smallest subspace containing .
Definition 4.5.17 (Direct sum of two subspaces). The sum is called direct, and written , if .
Proposition 4.5.18 (Characterizations of direct sums). For subspaces , the following are equivalent:
- is direct, i.e., ;
- every has a unique expression with , ;
- if with and , then .
Proof. (a) (b): Suppose . Then . The left side is in and the right side is in , so both lie in . Thus and .
(b) (c): If , then , and uniqueness gives and .
(c) (a): If , then with and . By (c), . ∎
Definition 4.5.19 (Direct sum decomposition). Subspaces of are independent if with implies for all . We say is the (internal) direct sum of , written if the are independent and . Equivalently, every can be written uniquely as with .
The equivalence is proved as in the case of two subspaces: two expressions give , and conversely uniqueness of the expression for is independence. For two subspaces, means and .
Note. For three or more subspaces, pairwise trivial intersections do not imply independence. In , the three lines , and meet pairwise only in , but .
Example 4.5.20.
- In , let (the -axis) and (the -plane). Then .
- In , fix , let consist of the vectors whose last entries are , and let consist of those whose first entries are . Then .
- Let be the vector space of arrays of elements of with entrywise operations (that is, for ; we study such matrices in Chapter 5), and let be the transpose of . Suppose . The symmetric matrices and the antisymmetric matrices are subspaces with . Indeed, every is with the first term in and the second in , and if then , so and . This fails in characteristic , where does not exist and , so that .
Note. Direct sums appear throughout linear algebra. Diagonalizing a linear operator means decomposing the space as a direct sum of eigenspaces, and if then the quotient space is isomorphic to ; both ideas are developed in Chapter 7. A direct sum decomposition lets us break a problem into independent pieces.
4.5.5 External Direct Sums and Products
We can also build new vector spaces from old ones by taking products.
Definition 4.5.21 (Direct product (external direct sum)). Given vector spaces over the same field , their direct product (or external direct sum) is with componentwise operations: It is also denoted or .
Note. The external and internal direct sums are related as follows. If , then the map is linear, surjective because , and injective because expressions are unique; so it is an isomorphism. Conversely, contains the subspaces and , and it is their internal direct sum. The same holds for summands.
Example 4.5.22. The space is isomorphic to via . This is why we freely identify with .
For an infinite family of vector spaces , the two constructions differ.
Definition 4.5.23 (Infinite direct sums and products). Let be a family of vector spaces over . Their direct product and direct sum are with componentwise operations; the direct sum is a subspace of the direct product. For finite the two coincide.
Example 4.5.24. Taking for every , we get , while is the space of eventually zero sequences, isomorphic to (Example 4.5.14).