# Notes on Linear Algebra

Last Modified 2022-03-15

**Definition: Vector Space, Vector, Scalar**

Let * vector space* over

- a set
$V$ , called the;**underlying set** - a distinguished element
$\vo \in V$ , called the;**zero vector** - a unary operation
$-_V: V \to V$ , written as$\vv \mapsto -\vv$ , called;**vector negation** - a binary operation
$+_V: V \times V \to V$ , written as$(\vv, \vw) \mapsto \vv + \vw$ , called;**vector addition** - a binary operation
$\cdot_V: F \times V \to V$ , written as$(a, \vv) \mapsto a \cdot \vv$ , called;**scalar multiplication**

satisfying the following requirements:

:**Additive structure**$\alg{V; \vo, -_V, +_V}$ is an abelian group.:**Identity**$1 \cdot \vv = \vv$ for all$\vv \in V$ .:**Multiplicative compatibility**$(a \cdot b) \cdot \vv = a \cdot (b \cdot \vv)$ for all$a, b \in F$ and$\vv \in V$ .:**Left distributivity**$a \cdot (\vv + \vw) = (a \cdot \vv) + (a \cdot \vw)$ for all$a \in F$ and$\vv, \vw \in V$ .:**Right distributivity**$(a + b) \cdot \vv = (a \cdot \vv) + (b \cdot \vv)$ for all$a, b \in F$ and$\vv \in V$ .

The elements of * vectors*, and the elements of

*.*

**scalars**In these notes, we will use boldface upright letters, such as

Adopting this convention also provides another benefit: namely, we no longer need to distinguish between the operations of scalar addition

The same goes for the two multiplication operations

As with groups, rings, and fields, it is common to denote a vector space

**Definition: Linear Combination**

Let * linear combination* of a finite sequence of vectors

where * coefficients* of the linear combination. For

**Definition: Span**

Let * span* of a subset

Because we defined *any* subset

**Definition: Linear Subspace**

Let * linear subspace* of

Note that the empty set *not* a linear subspace! The requirement of closure under linear combinations includes the *empty* linear combination, which the empty set fails to satisfy, since

**Pairwise Linear Combinations Suffice**

**Theorem:** Let

:**Contains the zero vector**$\vo \in S$ .: For all**Closed under pairwise linear combinations**$a, b \in F$ and$\vv, \vw \in S$ , we have$a\vv + b\vw \in S$ .

*Proof sketch:* We prove by induction on

**Definition: Affine Combination**

Let * affine combination* of a finite sequence of vectors

Note that, unlike linear combinations, there is no such thing as an empty affine combination. The sum of an empty sequence of scalars is

**Definition: Affine Span, Affine Hull**

Let * affine span* or

*of a subset*

**affine hull****Definition: Affine Set**

Let * affine set* in

Because there are no empty affine combinations, the empty set *does* (vacuously) qualify as an affine set.

**Definition: Conic Combination**

Let * conic combination* of a finite sequence of vectors

**Definition: Conic Hull**

Let * conic hull* of a set

**Definition: Cone**

Let * cone* in

Note that a cone is not necessarily closed under taking conic combinations!

**Definition: Convex Combination**

Let * convex combination* of a finite sequence of vectors

**Definition: Convex Hull**

Let * convex hull* of a set

**Definition: Convex Set**

Let * convex set* in

**Definition: Real Vector Space, Complex Vector Space**

A * real vector space* is a vector space over the field

*is a vector space over the field*

**complex vector space**