Linear algebra
> section 1.1 basic vectors
def: a vector is a quantity that has both magnitude & direction
↳ geometrically, a vector is a directed line segment (arrow) identified by an initial point
e .
.
g ya
<b (2 2) ,
* we can use vectors to de
i B <1 2) displacements/velocity/forc
"
= =
,
9 X
991 , 0
def: given a vector with an initial point a = (a1,a2,a3) & a terminal point b = (b1,b2,b3), we
↑ = B = <D , -
a ,, bz-azibs-az) in IR3
note : V = b -a , ,, Dz-az bz-Az , , .... bn-an> in IRh
def: a vector in n dimensions is an ordered list of n elements V =
V ,, Vz , ..., Vn) i
def: two vectors are equivalent if they have the same components
let i (a
=
, b) & v = < ,
d>
in IR2 note: true in n dimensi
= = v ) a=c & b = d
↳ equivalent vectors have the same length & direction
def: a vector is in standard position (s.p.) if its initial position is located at the origin
ya
<P(x y) ,
T = OP
#
>X
10 , 0
def of vector addition: if vector u & vector v are positioned so that the initial point of v is t
then the sum of u+v is the vector formed from the initial point of u to the terminal point o
Y
A
in 2D
T
1 7 note: called parallelogram law as
,def of scalar multiplication: if c is a scalar & v is a vector then the scalar multiple, cv, is the
the length of v
> if c>0 th'en cv has the same direction as v
> if c<0 then cv has the opposite direction as v
def of vector subtraction: its the same as addition but one of the vectors is multiplied by -
(opposite direc
def: the zero vector has a length of O & no specific direction
same length, opposite direction
properties of vectors: suppose u,v&w are vectors in the n dimension & s is a real number (s
properties hold
1. commutativity of vector addition
2. associability of vector addition
,proof of property 2:
theorem: if v is a vector in n dimensions & k is a scalar then:
def: suppose w, v1, v2,…, vk are vectors in the dimension n & c1, c2,…, ck are scalars then
the vectors v1, v2,…, vk if
can be written like thi
,
> section 1.1 basic vectors
def: a vector is a quantity that has both magnitude & direction
↳ geometrically, a vector is a directed line segment (arrow) identified by an initial point
e .
.
g ya
<b (2 2) ,
* we can use vectors to de
i B <1 2) displacements/velocity/forc
"
= =
,
9 X
991 , 0
def: given a vector with an initial point a = (a1,a2,a3) & a terminal point b = (b1,b2,b3), we
↑ = B = <D , -
a ,, bz-azibs-az) in IR3
note : V = b -a , ,, Dz-az bz-Az , , .... bn-an> in IRh
def: a vector in n dimensions is an ordered list of n elements V =
V ,, Vz , ..., Vn) i
def: two vectors are equivalent if they have the same components
let i (a
=
, b) & v = < ,
d>
in IR2 note: true in n dimensi
= = v ) a=c & b = d
↳ equivalent vectors have the same length & direction
def: a vector is in standard position (s.p.) if its initial position is located at the origin
ya
<P(x y) ,
T = OP
#
>X
10 , 0
def of vector addition: if vector u & vector v are positioned so that the initial point of v is t
then the sum of u+v is the vector formed from the initial point of u to the terminal point o
Y
A
in 2D
T
1 7 note: called parallelogram law as
,def of scalar multiplication: if c is a scalar & v is a vector then the scalar multiple, cv, is the
the length of v
> if c>0 th'en cv has the same direction as v
> if c<0 then cv has the opposite direction as v
def of vector subtraction: its the same as addition but one of the vectors is multiplied by -
(opposite direc
def: the zero vector has a length of O & no specific direction
same length, opposite direction
properties of vectors: suppose u,v&w are vectors in the n dimension & s is a real number (s
properties hold
1. commutativity of vector addition
2. associability of vector addition
,proof of property 2:
theorem: if v is a vector in n dimensions & k is a scalar then:
def: suppose w, v1, v2,…, vk are vectors in the dimension n & c1, c2,…, ck are scalars then
the vectors v1, v2,…, vk if
can be written like thi
,
