Write and Explain general properties of vector spaces and prove that
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Definition of a Vector Space
We have seen that vectors in Rn enjoy a collection of properties such as commutative, associative, and distributive properties. Other mathematical objects such as matrices and polynomials share the same properties. Instead of proving theorems separately for each of these objects, it is convenient to give a single proof for anything that has these properties. Below is a definition that collects some of the most common properties.
Definition
A vector space V is a set with two operations + and * that satisfy the following properties.
a. If u and v are elements of V, then u + vis and element of V (closure under +)
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 -uwith
u + (-u) = 0
b. If u is in V and c is a real number thenc*u is in V (closure under *)
c * (u + v) = c * u + c * v
(c + d) * u = c * u + d * u
c * (d * u) = (cd) * u
1 * u = u
You will recognize these properties as properties of vectors in Rn, however there is a large class of vector spaces that do not look at all like Rn.
Examples of Vector Spaces
Example P2
Consider the set P2 of polynomials of degree less than or equal to 2. Define + to be polynomial addition
(a1t2 + b1t + c2) + (a2t2 + b2t + c2) = (a1 + a2)t2 + (b1 + b2)t + (c1 + c2)
and * is defined by
k * (at2 + bt + c) = (ak)t2 + (kb)t + (kc)
This is a vector space. Most of the properties clearly hold. We will demonstrate a few of the properties. For example the 0 vector is the zero polynomial (0). We have
(at2 + bt + c) + 0 = 0 + (at2 + bt + c) = at2 + bt + c
Property b2 holds since
(r + s) * (at2 + bt + c) = (r + s)at2 + (r + s)bt + (r + s) c
= (ra + sa)t2 + (rb + sb)t + (rc + sc) = (ra)t2 + (sa)t2 + (rb)t + (sb)t + rc + sc
= (ra)t2 + (rb)t + rc + (sa)t2+ (sb)t + sc = r(at2 + bt + c) + s(at2 + bt + c)
We have seen that vectors in Rn enjoy a collection of properties such as commutative, associative, and distributive properties. Other mathematical objects such as matrices and polynomials share the same properties. Instead of proving theorems separately for each of these objects, it is convenient to give a single proof for anything that has these properties. Below is a definition that collects some of the most common properties.
Definition
A vector space V is a set with two operations + and * that satisfy the following properties.
a. If u and v are elements of V, then u + vis and element of V (closure under +)
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 -uwith
u + (-u) = 0
b. If u is in V and c is a real number thenc*u is in V (closure under *)
c * (u + v) = c * u + c * v
(c + d) * u = c * u + d * u
c * (d * u) = (cd) * u
1 * u = u
You will recognize these properties as properties of vectors in Rn, however there is a large class of vector spaces that do not look at all like Rn.
Examples of Vector Spaces
Example P2
Consider the set P2 of polynomials of degree less than or equal to 2. Define + to be polynomial addition
(a1t2 + b1t + c2) + (a2t2 + b2t + c2) = (a1 + a2)t2 + (b1 + b2)t + (c1 + c2)
and * is defined by
k * (at2 + bt + c) = (ak)t2 + (kb)t + (kc)
This is a vector space. Most of the properties clearly hold. We will demonstrate a few of the properties. For example the 0 vector is the zero polynomial (0). We have
(at2 + bt + c) + 0 = 0 + (at2 + bt + c) = at2 + bt + c
Property b2 holds since
(r + s) * (at2 + bt + c) = (r + s)at2 + (r + s)bt + (r + s) c
= (ra + sa)t2 + (rb + sb)t + (rc + sc) = (ra)t2 + (sa)t2 + (rb)t + (sb)t + rc + sc
= (ra)t2 + (rb)t + rc + (sa)t2+ (sb)t + sc = r(at2 + bt + c) + s(at2 + bt + c)
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