Determine whether the set of functions is a vector space

In summary, the given problem asks to determine which of the sets of continuous functions defined on a closed interval I = [a,b] are vector spaces under the operations of vector addition and scalar multiplication. The properties that need to be satisfied for a set to be a vector space are closure under addition and scalar multiplication, commutative and associative laws for addition, existence of an additive identity and additive inverses, and distributive and unitary laws for scalar multiplication.
  • #1
zabumafu
13
0

Homework Statement



Let I = [a,b], a closed interval. With addition and scalar multiplication as defined for all real-valued continuous functions defined on I, determine which of the following sets of functions is a vector space.

a) All continuous functions, f, such that f(a)=f(b)

b) All continuous functions, f, such that f(a)=f(b)=0

c) All continuous functions, f, such that f(a)=f(b)=-1

d) All continuous functions, f, such that f((a+b)/2)=0

e) All constant functions on [a,b]


Homework Equations



The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum u + v belongs to V.
(1) Commutative law: For all vectors u and v in V, u + v = v + u
(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w
(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V, 0 + v = v and v + 0 = v.
(4) Additive inverses: For each vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V, called an additive inverse of v, and denoted by - v.
The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:
Closure: If v in any vector in V, and c is any real number, then the product c · v belongs to V.
(5) Distributive law: For all real numbers c and all vectors u, v in V, c · (u + v) = c · u + c · v
(6) Distributive law: For all real numbers c, d and all vectors v in V, (c+d) · v = c · v + d · v
(7) Associative law: For all real numbers c,d and all vectors v in V, c · (d · v) = (cd) · v
(8) Unitary law: For all vectors v in V, 1 · v = v

The Attempt at a Solution



All I can think of is showing whether or not each function satisfies all those properties. The properties were not given but this is what I found on the internet. Our teacher told us we should have learned this back in calculus however I placed out of the classes at my school due to AP's and I've never seen this. Its not in my book and we don't have notes because it was suppose to "jog our memory". If someone could just show me how to determine if one of those problems is a vector field so I can figure out the rest that would be great!
 
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  • #2
zabumafu said:

Homework Statement



Let I = [a,b], a closed interval. With addition and scalar multiplication as defined for all real-valued continuous functions defined on I, determine which of the following sets of functions is a vector space.

a) All continuous functions, f, such that f(a)=f(b)

b) All continuous functions, f, such that f(a)=f(b)=0

c) All continuous functions, f, such that f(a)=f(b)=-1

d) All continuous functions, f, such that f((a+b)/2)=0

e) All constant functions on [a,b]

Homework Equations



The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum u + v belongs to V.
(1) Commutative law: For all vectors u and v in V, u + v = v + u
(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w
(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V, 0 + v = v and v + 0 = v.
(4) Additive inverses: For each vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V, called an additive inverse of v, and denoted by - v.
The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:
Closure: If v in any vector in V, and c is any real number, then the product c · v belongs to V.
(5) Distributive law: For all real numbers c and all vectors u, v in V, c · (u + v) = c · u + c · v
(6) Distributive law: For all real numbers c, d and all vectors v in V, (c+d) · v = c · v + d · v
(7) Associative law: For all real numbers c,d and all vectors v in V, c · (d · v) = (cd) · v
(8) Unitary law: For all vectors v in V, 1 · v = v

The Attempt at a Solution



All I can think of is showing whether or not each function satisfies all those properties. The properties were not given but this is what I found on the internet. Our teacher told us we should have learned this back in calculus however I placed out of the classes at my school due to AP's and I've never seen this. Its not in my book and we don't have notes because it was suppose to "jog our memory". If someone could just show me how to determine if one of those problems is a vector field so I can figure out the rest that would be great!

You don't have to worry about most of those properties. They are known to be true for numbers or functions. Just concentrate on the closure properties. You want to show that if f(x) and g(x) are in the set and a is a scalar then f(x)+g(x) is in the set and af(x) is in the set. Start with the first one. If f(a)=f(b) and g(a)=g(b) is it true that f(a)+g(a)=f(b)+g(b)?
 
  • #3
Okay so for a) if f(a)=f(b) and g(a)=g(b) then f(a)+g(a)=f(b)+g(b) so a is a vector space?
Then it'd be the same for b) since f(a)=f(b)=0 then g(a)=g(b)=0 so f(a)+g(a)=f(b)+g(b)=0 and therefore b is a vector space also?

then for c it shouldn't be a vector space since f(a)=f(b)=-1 and if g(a)=g(b)=-1 then f(a)+g(a) =f(b)+g(b)=-2 which isn't -1 so its not a vector space?
 
  • #4
zabumafu said:
Okay so for a) if f(a)=f(b) and g(a)=g(b) then f(a)+g(a)=f(b)+g(b) so a is a vector space?
Then it'd be the same for b) since f(a)=f(b)=0 then g(a)=g(b)=0 so f(a)+g(a)=f(b)+g(b)=0 and therefore b is a vector space also?

then for c it shouldn't be a vector space since f(a)=f(b)=-1 and if g(a)=g(b)=-1 then f(a)+g(a) =f(b)+g(b)=-2 which isn't -1 so its not a vector space?

You need to check closure under the scalar product as well, but yes, you are on the right track. And you should probably note that if f is continuous and g is continuous then f+g and af are also continuous. a) and b) are vector spaces. c) is not. Continue.
 
  • #5
To clarify what Dick said, to be a vector space, it must satisfy both closure of the sum and closure of the scalar product. For (a) and (b) you have shown the first but not yet the second. For (c), since it does not have closure of the sum of vectors, it doesn't matter whether the scalar product is closed and you do NOT have to look at that.
 

Related to Determine whether the set of functions is a vector space

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations that can be performed on those vectors. These operations include vector addition and scalar multiplication. A set of functions is considered a vector space if it follows certain properties, such as closure under addition and scalar multiplication.

2. What are the properties that determine whether a set of functions is a vector space?

The properties that determine whether a set of functions is a vector space include closure under addition and scalar multiplication, associativity and commutativity of addition, existence of an identity element for addition, existence of inverse elements for addition, distributivity of scalar multiplication over addition, and compatibility of scalar multiplication with field multiplication.

3. How do you check if a set of functions satisfies the properties of a vector space?

To check if a set of functions satisfies the properties of a vector space, you need to verify that the functions in the set follow the properties mentioned in question 2. This can be done by performing mathematical operations on the functions and checking if they still belong to the set and if the properties hold true.

4. What are some examples of sets of functions that are vector spaces?

Some examples of sets of functions that are vector spaces include the set of all polynomials of a certain degree, the set of all continuous functions on a given interval, and the set of all differentiable functions on a given interval. These sets follow the properties of a vector space and are commonly used in mathematical applications.

5. Are all sets of functions vector spaces?

No, not all sets of functions are vector spaces. A set of functions can only be considered a vector space if it satisfies the properties mentioned in question 2. If one or more of these properties do not hold true for a set of functions, then it cannot be considered a vector space. It is important to verify these properties before determining if a set of functions is a vector space.

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