This document provides notes on vector spaces, which are fundamental objects in linear algebra. It begins with examples of vector spaces such as R2, R3, C2, C3 and defines vector spaces more generally as sets that are closed under vector addition and scalar multiplication and satisfy other properties like the existence of additive identities. It then provides several examples of vector spaces including the set of all n-tuples over a field, the set of all m×n matrices, the set of differentiable functions on an interval, and the set of polynomials with coefficients in a field.

1. Vector Spaces• January 2018 • Linear Algebra notes
Vector Spaces
Quasar Chunawala, Mumbai
January 2018
Abstract
These are my notes on vector spaces, one of the most fundamental objects in linear algebra and the
whole of mathematics.
Contents
1 Vector spaces 1
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Conclusion 8
1. Vector spaces
1.1. Preliminaries
Let’s refresh what R2, R3, C2, C3 mean.
The vector space R2, which you can think of as a plane, consists of all ordered pairs of real
numbers.
R2
= {(x, y) : x, y ∈ R}
The vector space R3, which you can think of as ordinary space, consists of all ordered triples
of real numbers.
R3
= {(x, y, z) : x, y, z ∈ R}
The vector space Rn, which you can think of as n-dimensional space, consists of all possible
ordered lists of length n of real numbers. Such an ordered collection of n elements is called n-tuple.
Rn is the set of all such n-tuples.
Rn
= {(x1, x2, . . . , xn) : x1, x2, . . . , xn ∈ R}
If n ≥ 4, we cannot easily visualize Rn as a physical object. The same problem arises if we
work with complex numbers. C1 can be thought of as a plane. But, consider C2 defined as,
C2
= {(z1, z2) : z1, z2 ∈ C}
For n ≥ 2, the human brain cannot provide geometric models of Cn. However, even if n is
large, we can treat them as geometric vectors and perform algebraic manipulations in Fn, as easily
as in R2 or R3, as we will study shortly.
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2. Vector Spaces• January 2018 • Linear Algebra notes
Often the mathematics of Fn becomes cleaner, if we use a single entity to denote an n-tuple,
without explicitly writing its co-ordinates. For example, suppose the rule of addition is defined on
Fn by adding elements coordinate wise.
(x1, x2, . . . , xn) + (y1, y2, . . . , yn) = (x1 + y1, x2 + y2, . . . , xn + yn)
It is convenient if we represent
x = (x1, x2, x3, . . . , xn)
and call it a vector.
Thus, commutative property of addition in Fn should be expressed as
x + y = y + x
We begin with the basic concept of linear algebra. For the definition that follows, we assume
that we are given a particular field F, the scalars to be used are elements of F.
1.2. Definition.
A vector space (or a linear space) is a set V of elements satisfying the following axioms -
1. There is a function called vector addition, that assign to every pair of elements x, y ∈ V, an
element x + y in V called the sum of x and y, such that :
(a) Addition is commutative.
x + y = y + x
(b) Addition is associative.
x + (y + z) = (x + y) + z
(c) Existence of a zero vector.
There exists in V, a unique vector 0, called the 0 vector such that:
x + 0 = x, ∀x ∈ V
(d) Existence of a negative element.
To every vector x in V, there corresponds a unique vector −x, called the negative of x,
such that
x + (−x) = 0
where 0 is the 0 vector.
2. There is a function called scalar multiplication, that assigns to every pair α ∈ F and x ∈ V,
α · x in V, called the scalar product of α and x, in such a way that :
(a) Multiplication by scalars is associative.
α(βx) = (αβ)x
(b) Existence of a unit scalar.
1x = x
3. Distributive properties.
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3. Vector Spaces• January 2018 • Linear Algebra notes
(a) Scalar multiplication distributes over addition.
α(x + y) = αx + αy
(b) Scalar addition distributes over multiplication.
(α + β)x = αx + βx
The relation between the vector space V and the underlying field F is usually described by
saying that V is a vector space over F. If F is the field of real numbers, V is called a real vector
space, similarly if F is Q or C, we speak of rational vector spaces or complex vector spaces.
This axiomatic definition opens up a whole new world of things we can call vectors now.
For example, n- tuples, m × n matrices, complex numbers, the solutions of a linear differential
equation, polynomials are all vectors. Vectors are not necessarily elements of R2 or geometric
vectors.
1.3.
1. Let V = R = F. Define addition and multiplication in the usual way, real numbers are
added and multiplied by scalars. As elements in V are real numbers, and the set of real
numbers forms a field, the elements V satisfy all axioms of a vector space. Thus R is a vector
space over R.
2. Let V = C = F. Addition and multiplication are to be ordinary complex number addition
and multiplication. Show that, C is a vector space over C.
Proof. We know that, if z1, z2 ∈ C, =⇒ z1 + z2 ∈ C. Therefore, V is closed under addition.
• As C is a field, addition of complex numbers is commutative.
z1 + z2 = z2 + z1
• As C is a field, addition of complex numbers is associative.
z1 + (z2 + z3) = (z1 + z2) + z3
• Existence of zero vector. C is field and contains the zero vector 0 := 0 + i0, such that
z + 0 = z
• Existence of negative element. If z = x + iy ∈ C, there exists a unique element
−z = −x − iy ∈ C, such that,
z + (−z) = (x + iy) + (−x − iy) = (x − x) + i(y − y) = 0
We also know that if α ∈ C, x ∈ C, αx ∈ C. Hence, V is closed with respect to scalar
multiplication.
• Existence of unit element. As C is a field, it contains a multiplicative identity 1. Thus,
1z = z
holds.
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4. Vector Spaces• January 2018 • Linear Algebra notes
• Multiplication is associative. This holds as multiplication in C is associative.
α(βz) = (αβ)z
We also verify the distributive properties.
• Scalar multiplication distributes well over vector addition.
α(z1 + z2) = (α1 + iα2)(x1 + x2 + i(y1 + y2)
= α1(x1 + x2) − α2(y1 + y2) + iα1(y1 + y2) + iα2(x1 + x2)
= (α1x1 − α2y1) + i(α2x1 + α1y1) + (α1x2 − α2y2) + i(α2x2 + α1y2)
= (α1 + iα2)(x1 + iy1) + (α1 + iα2)(x2 + iy2)
= αz1 + αz2
• On similar lines, addition distributes well over multiplication.
(α + β)z = αz + βz
Thus, C is a vector space over C.
In general, any field is a vector space over itself. F(F) is a vector space.
3. Let V = F2 be the set of all column vectors which have just two components(co-ordinates).
F2
= {
x1
x2
: x1, x2 ∈ F}
Let x, y ∈ F2. Then, x =
x1
x2
and y =
y1
y2
. And let α ∈ F.
Define addition function as:
x + y =
x1 + x2
y1 + y2
Define scalar multiplication as:
αx =
αx1
αx2
It is an easy exercise to prove that F2 is a vector space over F, with respect to addition and
scalar multiplication defined above.
4. All n- tuples of real numbers form the vector space Rn over the real numbers R.
5. All n-tuples of complex numbers form the vector space Cn over the complex numbers C.
It is an easy exercise to prove that Fn is a vector space over F with respect to addition and
scalar multiplication defined in the usual way.
6. Let V = Fm×n be the set of all m × n matrices.
Fm×n
=



A =





a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
am1 am2 . . . amn





: aij ∈ F



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5. Vector Spaces• January 2018 • Linear Algebra notes
Let x, y be vectors in Fm×n. We define vector addition as ordinary matrix addition element-
wise.
x + y =





x11 + y11 x12 + y12 . . . x1n + y1n
x21 + y21 x22 + y22 . . . x2n + y2n
...
...
xm1 + ym1 xm2 + ym2 . . . xmn + ymn





We define scalar multiplication as
αx =





αx11 αx12 . . . αx1n
αx21 αx22 . . . αx2n
...
...
αxm1 αxm2 . . . αxmn





All m × n matrices form the vector space Fm×n over the field F, with respect to addition and
scalar multiplcation defined above.
Convenient notation:
C[a, b] - The set of all real valued functions defined from [a, b] onto F.
Ck(a, b) - The set of all k-times continuously differentiable functions from (a, b) onto F.
7. Let C([a, b]) be the set of all real valued functions defined from [a, b] onto R.
Let p, q be two functions in C([a, b]). Define addition as a new function p + q that assigns to
each t ∈ [a, b], the value p(t) + q(t).
(p + q)(t) = p(t) + q(t)
Define multiplication as a new function αp that assigns to each t ∈ [a, b], the value αp(t).
It is an easy exercise to prove that C([a, b]) is a vector space over R.
8. Let Ck((a, b)) be the set of all real valued functions f with the property that
dk f
dtk is continuous
in (a, b). Define vector addition and scalar multiplication as before. It is an easy exercise to
prove that Ck((a, b)) is a vector space over R.
We can do a little bit more general. One looks at C∞((a, b)).
9. Let C∞((a, b)) be the set of all real valued functions that are infinitely many times dif-
ferentiable in the open interval (a, b). The same operations of vector addition and scalar
multiplication will tell you that this is a real vector space.
10. Collect all functions f defined from (a, b) onto R in the set
F((a, b)) = {f : (a, b) → R}
In this we look at the set
V = {f ∈ F :
b
a
f (t)dt exists}
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6. Vector Spaces• January 2018 • Linear Algebra notes
that is the set of all functions that are Riemann integrable. Then, with respect to usual
addition and scalar multiplication, we can show that V is a real vector space.
If f is Riemann-integrable and g is Riemann-integrable, then (f + g) is Riemann integrable.
If f is Riemann-integrable and α is a real number, then α f is Riemann-integrable.
Polynomial functions:
A function p : F → F is a polynomial with coefficients a0, a1, a2, . . . , an ∈ F of degree n, if to
each x ∈ F, p assigns the value p(x) where
p(x) = a0 + a1x + a2x2
+ . . . + anxn
11. Let Pn(R) be set of all polynomials in real variable x with real coefficients of degree not
exceeding n.
Let p, q be two polynomials in Pn(R). Define sum of two polynomials as a new polynomial
(p + q)(x).
(p + q)(x) = p(x) + q(x)
= p0x + p1x + p2x2
+ . . . + pnxn
+ q0x + q1x + q2x2
+ . . . + qnxn
= (p0 + q0) + (p1 + q1)x + (p2 + q2)x2
+ . . . + (pn + qn)xn
We can easily see, that P(R) is closed under polynomial addition defined in the above way.
Clearly, polynomial addition is associative, commutative. There exists a 0 polynomial, such
that 0 + p = 0. There is unique negative polynomial −p = −p0 − p1x − p2x2 − . . . − pnxn,
such that p + (−p) = 0.
If c ∈ R, define scalar multiplication as a new polynomial cp.
(cp)(x) = cp(x)
= cp0 + cp1x + . . . + cpnxn
If we set c = 1, 1 · p = p. Scalar multiplication is associative. Further, the distributive
properties are also satisfied.
With these definitions of addition and scalar multiplication, Pn(R) is vector space over R.
12. Let A be a m × n matrix with its entries as real numbers. That is, A ∈ Rm×n. Consider the
system of m linear equations
Ax = 0
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7. Vector Spaces• January 2018 • Linear Algebra notes
Remember A is a m × n matrix, x =





x1
x2
...
xn





. Collect the set of all solution vectors in a set V
which is a subset of Rn. Rn already has vector addition and scalar multiplication defined. It
can be shown that V is a vector space over R with respect to those operations.
V = {x ∈ Rn
: Ax = 0} ⊆ R
Proof. Let us first show that V is closed with respect to addition. If x, y ∈ V that x, y satisfy
Ax = 0 and Ay = 0, then we must prove that x + y ∈ V that is, x + y satisfied A(x + y) = 0.
We know that matrix multiplication is distributive.
A(x + y) = Ax + Ay
= 0 + 0 = 0
Thus, (x + y) satisfies A(x + y) = 0.
Similarly, if x ∈ V that is x satisfies Ax = 0, then αx ∈ V, that is αx satisfies A(αx) = 0. This
is true, as
A(αx) = α(Ax)
= α(0) = 0
13. Let us look at an ordinary differential equation of the form
F(x, y, y , . . . , y(n)
) = a0(x)
dny
dxn
+ a1(x)
dn−1y
dxn−1
+ an(x) = 0
A function f (x) is the solution of this differential equation if and only if,
F[x, f (x), f (x), . . . , f (n)
(x)] = 0
Collect all such functions f (x) in a set V.
V = {f : F[x, f (x), f (x), . . . , f (n)
(x)] = 0}
It is an easy exercise to prove that the solutions of a linear homogenous differential equation
form a vector space V over the field R.
14. Let V ⊆ R2 be defined as
V := {(x1, x2) : x2 = 5x1 and x1, x2 ∈ R}
These are the set of all points on the straight line y = 5x in a plane. We define vector addition
and scalar multiplication in V to be the usual operations in R2.
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8. Vector Spaces• January 2018 • Linear Algebra notes
x + y = (x1 + y1, x2 + y2)
αx = (αx1, αx2)
Show that V is a vector space.
Proof.
2. Conclusion
“I always thought something was fundamentally wrong with the universe” [? ]
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