Lecture Notes on Vector Spaces and Linear Transformations (COMP11120)

Vector Spaces and Linear Transformations

Vectors
Definition A vector is
simply an ordered n-tuple of real numbers VI , Va
, ...,
Un Examples :



2
V
, Va , ...,Un are called the components or entries of .


= = R
1




ERb
-

2 4
We has dimension a c E
say X n.
O

-

1

Vi
We write : E An
Va
V
R
=
-
is the set of all n-dimensional vectors called n
: ,


Un
dimensional real space .




Vector Operations
Vector Addition Multiplication by a scalar a ER Vector Substraction

V, qV,
a
WI Vi + wi
V2 Wa & V2
( w)
↓
Vatwa av V w =
V +
= =

t = -


+ w =
:
I
: :
in
:
Un Wn Vn + Wn XVn where -w is shorthand for 1-1) ef




Zero Vector in Ru Position Vectors in R2
A vector with n entries
equal to 0
. ↓ =
(Y) and w =
(w) as position vectors

Zero Vector
ya
O so(V , Val
8
8 =
In entries (W , Wa&
V
: q
W
8 -




E &
V X




Equality in Rh Every point (vi , va) is uniquely represented as a position vector.
Vectors are ordered tuples of numbers : (2) * (2)
Vector Let and w be victors in Rh General View of Vectors R2
Equality 1 in


Then1 =
w if
N *
y
V = Wi
, Va = Wa , ..., Un =
Wh V
-
4-




V =
Y as
free vector
3-
-vej -



2 -




------
*




i
1 - Vil =



M



= It's n

The components V, and
ve of 1 are viewed as displacements wrt .
the unit vectors i andj in the
coordinate system .




Laws for vectors in Rh

Let , V ,
we Rh L , X , we R" and a,B be any scalars (real numbers)
·
V + w =
W + V commutativity ·
x (V + w) = XV + xW distributivity
O
u + (k + w) = (u + 1) + w associativity ·
(x + B(v = xx + B

·
V + 0 =V & is the additive unit. ·
x(B1) =
(xB) mixed associativity
0 + k =
V · /k =
V multiply with 1

· + ( 1)
-
= 0 -

V is the additive inverse



(x) + V = Q Of Y · Ov =
0 multiply with scalar O

·
x0 =
0 multiply witha vector 3
·
(-Nk = (v) = x( x)
-




multiply with negatives