Introduction to vector space properties:
A vector space is a set V consists of two operations: Vector addition and scalar multiplication. These operations satisfy assured properties. The scalars are taken as of a field F; anywhere F stands any for the real numbers R otherwise the complex numbers C. Consider the set V over a field F, if specified an operation vector addition describe in V, indicated v + w for every v, w in V, and an operation scalar multiplication in V, indicated b * v for every v in V and b in F. Let us see about the vector space properties.
Having problem with Vector Sum keep reading my upcoming posts, i will try to help you.
Description - Vector Space Properties:
Consider the vector space V is a set then the two operations addition and multiplication that satisfy the following properties.
Consider the two elements u and v in vector space V, after that u + v is and element of V .
u + v = v + u
u + (v + w) = (u + v) + w
There is an element 0 in V such that
u + 0 = 0 + u = u
For every u in V there is an element -u with
u + (-u) = 0
If u is in V and b is a real number after that b*u is in V
b * (u + v) = b * u + b * v
(b + c) * u = b * u + c * u
b * (c * u) = (bd) * u
1 * u = u
Vector spaces contain numerous additional properties. Followings are the some of the most basic ones.
Consider V is a vector space subsequently
0 * u = 0 for all u in V
c * 0 = 0 for all scalars c
If cu = 0 then either c = 0 or u = 0
(-1) u = -u for all u in V
Between, if you have problem on these topics Dot Product Example, please browse expert math related websites for more help on Finding Components of a Vector.
Example Problems for Vector Space Properties:
Example 1 for vector space properties
If u is vector then prove (-1) u = -u
Proof:
(-1) u = -u
(-1) u + u = (-1) u + (1) u
(-1 + 1) u = (-1) u + (1) u
0 * u = (-1) u + u
(-1) u + u = 0
Thus, (-1) u = -u
Example 2 for vector space properties:
If a and b are vectors then prove a + (b - a) = b
Proof:
a + (b - a) = a + (b + (-1) a)
= 1a + ((-1) a + b)
= (1a + (-1) a) + b
= (1 + (-1)) a + b
= (0 * a) + b
= 0 + b
= b
Proved that a + (b - a) = b
My Previous Blog :- http://advancemath.blogspot.in/2012/10/how-to-solve-matrix-equality.html
A vector space is a set V consists of two operations: Vector addition and scalar multiplication. These operations satisfy assured properties. The scalars are taken as of a field F; anywhere F stands any for the real numbers R otherwise the complex numbers C. Consider the set V over a field F, if specified an operation vector addition describe in V, indicated v + w for every v, w in V, and an operation scalar multiplication in V, indicated b * v for every v in V and b in F. Let us see about the vector space properties.
Having problem with Vector Sum keep reading my upcoming posts, i will try to help you.
Description - Vector Space Properties:
Consider the vector space V is a set then the two operations addition and multiplication that satisfy the following properties.
Consider the two elements u and v in vector space V, after that u + v is and element of V .
u + v = v + u
u + (v + w) = (u + v) + w
There is an element 0 in V such that
u + 0 = 0 + u = u
For every u in V there is an element -u with
u + (-u) = 0
If u is in V and b is a real number after that b*u is in V
b * (u + v) = b * u + b * v
(b + c) * u = b * u + c * u
b * (c * u) = (bd) * u
1 * u = u
Vector spaces contain numerous additional properties. Followings are the some of the most basic ones.
Consider V is a vector space subsequently
0 * u = 0 for all u in V
c * 0 = 0 for all scalars c
If cu = 0 then either c = 0 or u = 0
(-1) u = -u for all u in V
Between, if you have problem on these topics Dot Product Example, please browse expert math related websites for more help on Finding Components of a Vector.
Example Problems for Vector Space Properties:
Example 1 for vector space properties
If u is vector then prove (-1) u = -u
Proof:
(-1) u = -u
(-1) u + u = (-1) u + (1) u
(-1 + 1) u = (-1) u + (1) u
0 * u = (-1) u + u
(-1) u + u = 0
Thus, (-1) u = -u
Example 2 for vector space properties:
If a and b are vectors then prove a + (b - a) = b
Proof:
a + (b - a) = a + (b + (-1) a)
= 1a + ((-1) a + b)
= (1a + (-1) a) + b
= (1 + (-1)) a + b
= (0 * a) + b
= 0 + b
= b
Proved that a + (b - a) = b
My Previous Blog :- http://advancemath.blogspot.in/2012/10/how-to-solve-matrix-equality.html
No comments:
Post a Comment