Now that you are familiar with vector spaces, try determining if other sets of objects meet the requirements (or not). For example, the set of integers, the set of rationals, the set of positive continuous functions on a closed interval.
Do these meet the definition of a vector space? If not, why?
Can you think of other examples from your past?
Thank you.
Solution
set of integers are not vectors space .......becaue .
take any integer say ; 6.
now take any real number say; 3/5
but; 3/5*6 = 18/5 .......but this is not an integer .so ; it is not closed under scalar multiplication .
so; set of integers are not a vector space
set of ; rational are also not vector space .
because;
take any a rational ......2
now you take a real number (scalar)Â Â Â Â Â Â squreroot(2) .
now; swqurt(2) *2 = an irrational number .
so; rational are not alosed under scalar multiplication.
positive continuos functions are not vector space under addition because ......in this case there will no inverse of any given function.also;
positive conctinuos functions are not vector space under multiplication:)
because if .......  f and a is any negative real number then
a*f = is a negative continuos function :)
again it is not alosed under scalar field.
now you can construct a vector space .
it is ........
set of all 3 cross 3 matries which are invertioble .......under matrix multiplication .
good luck!!!!!!!!!!!!!
.
The byproduct of sericulture in different industries.pptx
Now that you are familiar with vector spaces- try determining if other.docx
1. Now that you are familiar with vector spaces, try determining if other sets of objects meet the
requirements (or not). For example, the set of integers, the set of rationals, the set of positive
continuous functions on a closed interval.
Do these meet the definition of a vector space? If not, why?
Can you think of other examples from your past?
Thank you.
Solution
set of integers are not vectors space .......becaue .
take any integer say ; 6.
now take any real number say; 3/5
but; 3/5*6 = 18/5 .......but this is not an integer .so ; it is not closed under scalar multiplication .
so; set of integers are not a vector space
set of ; rational are also not vector space .
because;
take any a rational ......2
now you take a real number (scalar)Â Â Â Â Â Â squreroot(2) .
now; swqurt(2) *2 = an irrational number .
so; rational are not alosed under scalar multiplication.
positive continuos functions are not vector space under addition because ......in this case there
will no inverse of any given function.also;
positive conctinuos functions are not vector space under multiplication:)
because if .......  f and a is any negative real number then
a*f = is a negative continuos function :)
2. again it is not alosed under scalar field.
now you can construct a vector space .
it is ........
set of all 3 cross 3 matries which are invertioble .......under matrix multiplication .
good luck!!!!!!!!!!!!!
