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Identifying subspaces (Linear Algebra)
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I want to identify all the types of subspaces of a set of functions from R to R.
I believe
- the set of all polynomials
- the set of non-negative continuous functions
- the set of constant functions
are all considered subspaces. However,I am unsure if the (set of functions that go through the origin) are considered subspaces since I am unsure of how to close that under multiplication.
linear-algebra vector-spaces linear-transformations
asked Jul 22 at 9:06
DoofusAnarchy
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up vote
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down vote
favorite
I want to identify all the types of subspaces of a set of functions from R to R.
I believe
- the set of all polynomials
- the set of non-negative continuous functions
- the set of constant functions
are all considered subspaces. However,I am unsure if the (set of functions that go through the origin) are considered subspaces since I am unsure of how to close that under multiplication.
linear-algebra vector-spaces linear-transformations
asked Jul 22 at 9:06
DoofusAnarchy
74
1
Why do you think the second one is a subspace? And for your question, what do you know about a function that satisfies $f(0)=0$ if you multiply it with a scalar or add the same type of function?
– The Phenotype
Jul 22 at 9:26
Ahhh. I understand my mistake. So the second is not a subspace because it can be empty (zero vector) since it's non-negative. Likewise, the set of functions going through the origin can also be empty. So the subspaces are (the set of all polynomials, and the set of constant functions)
– DoofusAnarchy
Jul 22 at 9:34
Identifying all the types of subspaces really says nothing. there are infinitely many different subspaces, so identifying all of them is impossible, and the definition of "types" is not clear. After that, you don't really have any question here. If you're just looking for examples, then the polynomials are a good example of a vector space of finite dimension and you can make from polynomials more elaborate examples for example: the space of polynomials of degree 3 but with the first degree zero (i.e spanned by $x^3,x^2,1$)
– GuySa
Jul 22 at 9:45
@DoofusAnarchy I don't understand what you mean by "it can be empty", as the subset of non-negative functions taken from the set of all functions (assuming this space is the given set, else the question is completely dependent on the given set of functions and hence makes no sense) is definitely non-empty as it contains at least the constant function $1$. The correct reasoning is that multiplication with the scalar $-1$ makes it not a subspace. Your fourth one is also a subspace.
– The Phenotype
Jul 22 at 9:45
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I want to identify all the types of subspaces of a set of functions from R to R.
I believe
- the set of all polynomials
- the set of non-negative continuous functions
- the set of constant functions
are all considered subspaces. However,I am unsure if the (set of functions that go through the origin) are considered subspaces since I am unsure of how to close that under multiplication.
linear-algebra vector-spaces linear-transformations
asked Jul 22 at 9:06
DoofusAnarchy
74
I want to identify all the types of subspaces of a set of functions from R to R.
I believe
- the set of all polynomials
- the set of non-negative continuous functions
- the set of constant functions
are all considered subspaces. However,I am unsure if the (set of functions that go through the origin) are considered subspaces since I am unsure of how to close that under multiplication.
linear-algebra vector-spaces linear-transformations
asked Jul 22 at 9:06
DoofusAnarchy
74
asked Jul 22 at 9:06
DoofusAnarchy
74
asked Jul 22 at 9:06
DoofusAnarchy
74
asked Jul 22 at 9:06
DoofusAnarchy
74
74
1
Why do you think the second one is a subspace? And for your question, what do you know about a function that satisfies $f(0)=0$ if you multiply it with a scalar or add the same type of function?
– The Phenotype
Jul 22 at 9:26
Ahhh. I understand my mistake. So the second is not a subspace because it can be empty (zero vector) since it's non-negative. Likewise, the set of functions going through the origin can also be empty. So the subspaces are (the set of all polynomials, and the set of constant functions)
– DoofusAnarchy
Jul 22 at 9:34
Identifying all the types of subspaces really says nothing. there are infinitely many different subspaces, so identifying all of them is impossible, and the definition of "types" is not clear. After that, you don't really have any question here. If you're just looking for examples, then the polynomials are a good example of a vector space of finite dimension and you can make from polynomials more elaborate examples for example: the space of polynomials of degree 3 but with the first degree zero (i.e spanned by $x^3,x^2,1$)
– GuySa
Jul 22 at 9:45
@DoofusAnarchy I don't understand what you mean by "it can be empty", as the subset of non-negative functions taken from the set of all functions (assuming this space is the given set, else the question is completely dependent on the given set of functions and hence makes no sense) is definitely non-empty as it contains at least the constant function $1$. The correct reasoning is that multiplication with the scalar $-1$ makes it not a subspace. Your fourth one is also a subspace.
– The Phenotype
Jul 22 at 9:45
add a comment |Â
1
Why do you think the second one is a subspace? And for your question, what do you know about a function that satisfies $f(0)=0$ if you multiply it with a scalar or add the same type of function?
– The Phenotype
Jul 22 at 9:26
Ahhh. I understand my mistake. So the second is not a subspace because it can be empty (zero vector) since it's non-negative. Likewise, the set of functions going through the origin can also be empty. So the subspaces are (the set of all polynomials, and the set of constant functions)
– DoofusAnarchy
Jul 22 at 9:34
Identifying all the types of subspaces really says nothing. there are infinitely many different subspaces, so identifying all of them is impossible, and the definition of "types" is not clear. After that, you don't really have any question here. If you're just looking for examples, then the polynomials are a good example of a vector space of finite dimension and you can make from polynomials more elaborate examples for example: the space of polynomials of degree 3 but with the first degree zero (i.e spanned by $x^3,x^2,1$)
– GuySa
Jul 22 at 9:45
@DoofusAnarchy I don't understand what you mean by "it can be empty", as the subset of non-negative functions taken from the set of all functions (assuming this space is the given set, else the question is completely dependent on the given set of functions and hence makes no sense) is definitely non-empty as it contains at least the constant function $1$. The correct reasoning is that multiplication with the scalar $-1$ makes it not a subspace. Your fourth one is also a subspace.
– The Phenotype
Jul 22 at 9:45
1
1
Why do you think the second one is a subspace? And for your question, what do you know about a function that satisfies $f(0)=0$ if you multiply it with a scalar or add the same type of function?
– The Phenotype
Jul 22 at 9:26
Why do you think the second one is a subspace? And for your question, what do you know about a function that satisfies $f(0)=0$ if you multiply it with a scalar or add the same type of function?
– The Phenotype
Jul 22 at 9:26
Ahhh. I understand my mistake. So the second is not a subspace because it can be empty (zero vector) since it's non-negative. Likewise, the set of functions going through the origin can also be empty. So the subspaces are (the set of all polynomials, and the set of constant functions)
– DoofusAnarchy
Jul 22 at 9:34
Ahhh. I understand my mistake. So the second is not a subspace because it can be empty (zero vector) since it's non-negative. Likewise, the set of functions going through the origin can also be empty. So the subspaces are (the set of all polynomials, and the set of constant functions)
– DoofusAnarchy
Jul 22 at 9:34
Identifying all the types of subspaces really says nothing. there are infinitely many different subspaces, so identifying all of them is impossible, and the definition of "types" is not clear. After that, you don't really have any question here. If you're just looking for examples, then the polynomials are a good example of a vector space of finite dimension and you can make from polynomials more elaborate examples for example: the space of polynomials of degree 3 but with the first degree zero (i.e spanned by $x^3,x^2,1$)
– GuySa
Jul 22 at 9:45
Identifying all the types of subspaces really says nothing. there are infinitely many different subspaces, so identifying all of them is impossible, and the definition of "types" is not clear. After that, you don't really have any question here. If you're just looking for examples, then the polynomials are a good example of a vector space of finite dimension and you can make from polynomials more elaborate examples for example: the space of polynomials of degree 3 but with the first degree zero (i.e spanned by $x^3,x^2,1$)
– GuySa
Jul 22 at 9:45
@DoofusAnarchy I don't understand what you mean by "it can be empty", as the subset of non-negative functions taken from the set of all functions (assuming this space is the given set, else the question is completely dependent on the given set of functions and hence makes no sense) is definitely non-empty as it contains at least the constant function $1$. The correct reasoning is that multiplication with the scalar $-1$ makes it not a subspace. Your fourth one is also a subspace.
– The Phenotype
Jul 22 at 9:45
@DoofusAnarchy I don't understand what you mean by "it can be empty", as the subset of non-negative functions taken from the set of all functions (assuming this space is the given set, else the question is completely dependent on the given set of functions and hence makes no sense) is definitely non-empty as it contains at least the constant function $1$. The correct reasoning is that multiplication with the scalar $-1$ makes it not a subspace. Your fourth one is also a subspace.
– The Phenotype
Jul 22 at 9:45
add a comment |Â
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1
Why do you think the second one is a subspace? And for your question, what do you know about a function that satisfies $f(0)=0$ if you multiply it with a scalar or add the same type of function?
– The Phenotype
Jul 22 at 9:26
Ahhh. I understand my mistake. So the second is not a subspace because it can be empty (zero vector) since it's non-negative. Likewise, the set of functions going through the origin can also be empty. So the subspaces are (the set of all polynomials, and the set of constant functions)
– DoofusAnarchy
Jul 22 at 9:34
Identifying all the types of subspaces really says nothing. there are infinitely many different subspaces, so identifying all of them is impossible, and the definition of "types" is not clear. After that, you don't really have any question here. If you're just looking for examples, then the polynomials are a good example of a vector space of finite dimension and you can make from polynomials more elaborate examples for example: the space of polynomials of degree 3 but with the first degree zero (i.e spanned by $x^3,x^2,1$)
– GuySa
Jul 22 at 9:45
@DoofusAnarchy I don't understand what you mean by "it can be empty", as the subset of non-negative functions taken from the set of all functions (assuming this space is the given set, else the question is completely dependent on the given set of functions and hence makes no sense) is definitely non-empty as it contains at least the constant function $1$. The correct reasoning is that multiplication with the scalar $-1$ makes it not a subspace. Your fourth one is also a subspace.
– The Phenotype
Jul 22 at 9:45