Linear Algebra
The power of abstraction
Akira Nonaka

Abstract vector
Vectors in vector spaces do not necessarily have to be
arrow-like objects . Vectors are regarded as abstract
mathematical objects with particular properties, which in
some cases can be visualized as arrows.

Vector space
• A vector space over a field F is a set V together with
two operations that satisfy the eight axioms listed
below.
• You can consider F is real number at this moment.

Two operations
1. The first operation, called vector addition or simply
addition + : V × V → V, takes any two vectors v and w and
assigns to them a third vector which is commonly written
as v + w, and called the sum of these two vectors. (Note
that the resultant vector is also an element of the set V ).
2. The second operation, called scalar multiplication · : F × V
→ V， takes any scalar a and any vector v and gives
another vector av. (Similarly, the vector av is an element
of the set V )

Axioms

Associativity of addition
u + (v + w) = (u + v) + w

Commutativity of addition
u + v = v + u

Identity element of addition
There exists an element 0 ∈
V, called the zero vector,
such that v + 0 = v for all v ∈
V.

Inverse elements of addition
For every v ∈ V, there exists
an element −v ∈ V, called the
additive inverse of v, such
that v + (−v) = 0.

Compatibility of scalar multiplication
with field multiplication
a(bv) = (ab)v

Identity element of scalar multiplication
1v = v, where 1 denotes the
multiplicative identity in F.

a(u + v) = au + av
Distributivity of scalar multiplication
with respect to vector addition

(a + b)v = av + bv
Distributivity of scalar multiplication
with respect to field addition

Eight axioms
Axiom Meaning
Associativity of addition u + (v + w) = (u + v) + w
Commutativity of addition u + v = v + u
Identity element of addition There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of addition
For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such
that v + (−v) = 0.
Compatibility of scalar multiplication with field
multiplication
a(bv) = (ab)v
Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F.
Distributivity of scalar multiplication with respect to
vector addition
a(u + v) = au + av
Distributivity of scalar multiplication with respect to field
addition
(a + b)v = av + bv

Examples of vector spaces
1. Coordinate spaces
2. Matrices
3. Polynomials
4. etc.

Concepts of vector space
1. Linear span
2. Dimension
3. Eigenvalues and Eigenvectors
4. Inner Product
5. and much more…

Linear map
1. V, W : Vector space,
2. f: Function V -> W
3. u, v ∈ V, c ∈ Filed F
4. f is said to be linear map if
5. f(u+v) = f(u) + f(v)
6. f(cv) = c f(v)

Question!
What is the zero vector of
the vector space of linear
maps?

Answer
g(v) = 0
because…
v, w ∈ V
f(v) + g(w) = f(v) + 0 = f(v)
g(v) is linear map

Set of all linear maps
1. Set of all linear maps from V to W forms another
vector space.
2. V, W : vector space
3. Linear map f: V -> W
4. Set of linear maps L(V,W) : vector space

Process of abstraction
1. Mathematicians are researching different areas of
mathematics.
2. They found there are similar structures behind areas
they are studying.
3. They distilled the similarity and built the abstract
algebra.

Power of abstraction
1. All the theorems of linear algebra are proven using
these axioms.
2. Invent your own vector space and check if it satisfies
these axioms.
3. Then all the proven liner algebra theorems can be
applied to your vector space.

Reference
1. https://en.wikipedia.org/wiki/Vector_space
2. https://en.wikipedia.org/wiki/Linear_map
3. https://youtu.be/ozwodzD5bJM