Math
I exam: 24 January
VECTORS
A vector v with no n-components is an ordered set of n numbers
il
i
ex
i
sx
file
Lianne
It
usa
i
dimension
ofvector
MATRIX
A matrix A of order n x m is a table of doubly ordered numbers, with n rows and m columns. We
denote with
, the element of A in the i-th row and in the j-th column
Ai
A
f
ans 3
922
1
921 a
A square matrix is a matrix of dimension n m, theft is: the number of rows is equal to the
number of columns
ex
a
A
SUM
Let A and B be 2 matrices of order n m, them the matrix C of order n m, such that
for i
in and
m
5 1,2
1,2
Ci ai this
NOTE:
ATB
The neutral element of the sum is the
null matrix of order n m, theft is the
ex
a
23
Bil
p
matrix of dimension n m whose
23
to
on
2 13 1
en
on
elements are all zero
Ie
1
• product by a constant (scalar) k
the product between a matrix A of order n m and a scalar k is a matrix of order n m and
elements
dij
e
ex
K
i 1,2 in
Te 1,2
m
p
I I
34
1
• transposition
let A E M (n m), then ATEM min is called transpose of A and it elements are
is simply obtained by exchanging the rows of A with its columns
At
ex
At
È
It
A
HAT KB
ha KB
BTAT
AB
• Inner product or scalar product
given 2 vectors v (n 1) and w (n 1), the inner product is the scalar
a
IT
13
Ait Ati
Ità
DTT Eh vieni
IT
Iv
It In
Va V3
VW
II
Wa Wa Wa
VaWat V3Ws
it I
WVa Wavatwab
(Row-column product)
ità
III
E
vini
2
2 vectors in Rn (that is with n-components) are called orthogonal if their inner product is the
scalar 0.
Thus, v and w are orthogonal if
0
Ità
Xi ER
I
for
it 1,2
X
X ER
h
X
ER
XNER
ieri
_È
lX
il
I
if
i d È ER
I
a. Are v and w orthogonal?
b. Are v and Z orthogonal?
a
ITL121
II Ù 2
b
8 10
1,2
IT È
2 2
IT
How to represent
Y
no
E
a
2
x
1
• product between matrices
Let A
such that
Ci
ex
È Aix by
Eb
, them their product is a matrix
i 1,2
m 5
1,2
P
0
YES
Note 1: by definition the product AB is feasible if and only if the number of columns of A is equal
to the number of rows of B, that is if and only if the columns of the left matrix are the same
number of the rows of the right matrix
Note 2: the product AB is a matrix of dimension m p that is C is a matrix with a number of rows
equal to the number of rows of A and with a number of columns equal to the number of the
same order
Note 3: in general AB
BA. A necessary, but not sufficient condition to have AB = BA is A and B
being square matrix of the same order
A B
ex
visible
Let A
B A
3 a 2.3
a
visible
E M (n m), then the matrix I E
M (n n), such that AI = IA = A
IDENTITY MATRIX
In
4