Mathematical Techniques for Computer Science (COMP11120)
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Lecture Notes on Vector Spaces and Linear Transformations (COMP11120)
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Course
Mathematical Techniques for Computer Science (COMP11120)
Institution
The University Of Manchester (UOM)
Master the fundamentals of vector spaces and linear transformations with these comprehensive lecture notes for COMP11120. These notes cover key topics such as vector spaces, subspaces, bases, dimension, linear transformations, matrix representations, eigenvalues, and eigenvectors. With clear explan...
Mathematical Techniques for Computer Science (COMP11120)
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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
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