More questions on quadratic forms over field












2












$begingroup$


I'm a student, currently studying about quadratic forms over a field $mathbb{R}$ and I have a few questions regarding the topic.




  1. From a book I currently read, a quadratic form is a real-valued function over a vector space $E$ (i.e. $Q: E longrightarrow mathbb{R}$, please correct me if I'm mistaken) such that there exists a symmetric bilinear form $B: E times E longrightarrow mathbb{R}$ in which the following expression is valid:
    begin{align}
    Q(x)=B(x,x)
    end{align}
    $forall x in E$.


My question: given some function $F: E longrightarrow mathbb{R}$. The definition requires the existence of a symmetric bilinear form $G: E times E longrightarrow mathbb{R}$ in which the above expression valid, but how to make sure that we could have such function? For example, if I have a function $F: mathbb{R^3} longrightarrow mathbb{R}$ such that
begin{align}
F(x)= x_1+x_2+x_3
end{align}
with $ x=(x_1,x_2,x_3) in mathbb{R^3}$,



how to check whether $F$ is a quadratic form or not?




  1. The book also mentioned a regular quadratic space $(E, Q)$, i.e. a vector space $E$ in which it has a quadratic space $Q$ that is nonsingular. What is the definition of nonsingular in terms of function/transformation? And how to connect it to this?


I'm really lost, I know I'm still learning, and I need lots of help. Of course, this is not the last time I'll ask, maybe I'll come again if I find more difficulties but for now, this is all I've got to ask you good people in this community. Thanks! Any help will do!






linear-algebra abstract-algebra quadratic-forms







share|cite|improve this question









$endgroup$












  • $begingroup$
    The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms.
    $endgroup$
    – amd
    Dec 2 '18 at 3:18










  • $begingroup$
    Show that for $E$ finite dimensional with basis $(e_i)$ then $q(sum_i x_i e_i) = sum_i sum_j frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^top frac{A+A^top}{2} y, q(x) = x^top frac{A+A^top}{2} x$
    $endgroup$
    – reuns
    Dec 2 '18 at 4:01












  • $begingroup$
    @amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then?
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:41










  • $begingroup$
    @reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:50










  • $begingroup$
    en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms
    $endgroup$
    – reuns
    Dec 2 '18 at 22:16
















2












$begingroup$


I'm a student, currently studying about quadratic forms over a field $mathbb{R}$ and I have a few questions regarding the topic.




  1. From a book I currently read, a quadratic form is a real-valued function over a vector space $E$ (i.e. $Q: E longrightarrow mathbb{R}$, please correct me if I'm mistaken) such that there exists a symmetric bilinear form $B: E times E longrightarrow mathbb{R}$ in which the following expression is valid:
    begin{align}
    Q(x)=B(x,x)
    end{align}
    $forall x in E$.


My question: given some function $F: E longrightarrow mathbb{R}$. The definition requires the existence of a symmetric bilinear form $G: E times E longrightarrow mathbb{R}$ in which the above expression valid, but how to make sure that we could have such function? For example, if I have a function $F: mathbb{R^3} longrightarrow mathbb{R}$ such that
begin{align}
F(x)= x_1+x_2+x_3
end{align}
with $ x=(x_1,x_2,x_3) in mathbb{R^3}$,



how to check whether $F$ is a quadratic form or not?




  1. The book also mentioned a regular quadratic space $(E, Q)$, i.e. a vector space $E$ in which it has a quadratic space $Q$ that is nonsingular. What is the definition of nonsingular in terms of function/transformation? And how to connect it to this?


I'm really lost, I know I'm still learning, and I need lots of help. Of course, this is not the last time I'll ask, maybe I'll come again if I find more difficulties but for now, this is all I've got to ask you good people in this community. Thanks! Any help will do!






linear-algebra abstract-algebra quadratic-forms







share|cite|improve this question









$endgroup$












  • $begingroup$
    The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms.
    $endgroup$
    – amd
    Dec 2 '18 at 3:18










  • $begingroup$
    Show that for $E$ finite dimensional with basis $(e_i)$ then $q(sum_i x_i e_i) = sum_i sum_j frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^top frac{A+A^top}{2} y, q(x) = x^top frac{A+A^top}{2} x$
    $endgroup$
    – reuns
    Dec 2 '18 at 4:01












  • $begingroup$
    @amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then?
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:41










  • $begingroup$
    @reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:50










  • $begingroup$
    en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms
    $endgroup$
    – reuns
    Dec 2 '18 at 22:16














2












2








2





$begingroup$


I'm a student, currently studying about quadratic forms over a field $mathbb{R}$ and I have a few questions regarding the topic.




  1. From a book I currently read, a quadratic form is a real-valued function over a vector space $E$ (i.e. $Q: E longrightarrow mathbb{R}$, please correct me if I'm mistaken) such that there exists a symmetric bilinear form $B: E times E longrightarrow mathbb{R}$ in which the following expression is valid:
    begin{align}
    Q(x)=B(x,x)
    end{align}
    $forall x in E$.


My question: given some function $F: E longrightarrow mathbb{R}$. The definition requires the existence of a symmetric bilinear form $G: E times E longrightarrow mathbb{R}$ in which the above expression valid, but how to make sure that we could have such function? For example, if I have a function $F: mathbb{R^3} longrightarrow mathbb{R}$ such that
begin{align}
F(x)= x_1+x_2+x_3
end{align}
with $ x=(x_1,x_2,x_3) in mathbb{R^3}$,



how to check whether $F$ is a quadratic form or not?




  1. The book also mentioned a regular quadratic space $(E, Q)$, i.e. a vector space $E$ in which it has a quadratic space $Q$ that is nonsingular. What is the definition of nonsingular in terms of function/transformation? And how to connect it to this?


I'm really lost, I know I'm still learning, and I need lots of help. Of course, this is not the last time I'll ask, maybe I'll come again if I find more difficulties but for now, this is all I've got to ask you good people in this community. Thanks! Any help will do!






linear-algebra abstract-algebra quadratic-forms







share|cite|improve this question









$endgroup$




I'm a student, currently studying about quadratic forms over a field $mathbb{R}$ and I have a few questions regarding the topic.




  1. From a book I currently read, a quadratic form is a real-valued function over a vector space $E$ (i.e. $Q: E longrightarrow mathbb{R}$, please correct me if I'm mistaken) such that there exists a symmetric bilinear form $B: E times E longrightarrow mathbb{R}$ in which the following expression is valid:
    begin{align}
    Q(x)=B(x,x)
    end{align}
    $forall x in E$.


My question: given some function $F: E longrightarrow mathbb{R}$. The definition requires the existence of a symmetric bilinear form $G: E times E longrightarrow mathbb{R}$ in which the above expression valid, but how to make sure that we could have such function? For example, if I have a function $F: mathbb{R^3} longrightarrow mathbb{R}$ such that
begin{align}
F(x)= x_1+x_2+x_3
end{align}
with $ x=(x_1,x_2,x_3) in mathbb{R^3}$,



how to check whether $F$ is a quadratic form or not?




  1. The book also mentioned a regular quadratic space $(E, Q)$, i.e. a vector space $E$ in which it has a quadratic space $Q$ that is nonsingular. What is the definition of nonsingular in terms of function/transformation? And how to connect it to this?


I'm really lost, I know I'm still learning, and I need lots of help. Of course, this is not the last time I'll ask, maybe I'll come again if I find more difficulties but for now, this is all I've got to ask you good people in this community. Thanks! Any help will do!






linear-algebra abstract-algebra quadratic-forms




linear-algebra abstract-algebra quadratic-forms






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 2 '18 at 3:10









20gobbledigook0820gobbledigook08

111




111












  • $begingroup$
    The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms.
    $endgroup$
    – amd
    Dec 2 '18 at 3:18










  • $begingroup$
    Show that for $E$ finite dimensional with basis $(e_i)$ then $q(sum_i x_i e_i) = sum_i sum_j frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^top frac{A+A^top}{2} y, q(x) = x^top frac{A+A^top}{2} x$
    $endgroup$
    – reuns
    Dec 2 '18 at 4:01












  • $begingroup$
    @amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then?
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:41










  • $begingroup$
    @reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:50










  • $begingroup$
    en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms
    $endgroup$
    – reuns
    Dec 2 '18 at 22:16


















  • $begingroup$
    The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms.
    $endgroup$
    – amd
    Dec 2 '18 at 3:18










  • $begingroup$
    Show that for $E$ finite dimensional with basis $(e_i)$ then $q(sum_i x_i e_i) = sum_i sum_j frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^top frac{A+A^top}{2} y, q(x) = x^top frac{A+A^top}{2} x$
    $endgroup$
    – reuns
    Dec 2 '18 at 4:01












  • $begingroup$
    @amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then?
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:41










  • $begingroup$
    @reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 13:50










  • $begingroup$
    en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms
    $endgroup$
    – reuns
    Dec 2 '18 at 22:16
















$begingroup$
The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms.
$endgroup$
– amd
Dec 2 '18 at 3:18




$begingroup$
The key word is “quadratic.” When expanded in terms of coordinates, the expression for $Q$ will involve only second-degree terms.
$endgroup$
– amd
Dec 2 '18 at 3:18












$begingroup$
Show that for $E$ finite dimensional with basis $(e_i)$ then $q(sum_i x_i e_i) = sum_i sum_j frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^top frac{A+A^top}{2} y, q(x) = x^top frac{A+A^top}{2} x$
$endgroup$
– reuns
Dec 2 '18 at 4:01






$begingroup$
Show that for $E$ finite dimensional with basis $(e_i)$ then $q(sum_i x_i e_i) = sum_i sum_j frac{A_{i,j}+A_{j,i}}{2} x_i x_j$. In matrix form $B(x,y) = x^top frac{A+A^top}{2} y, q(x) = x^top frac{A+A^top}{2} x$
$endgroup$
– reuns
Dec 2 '18 at 4:01














$begingroup$
@amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then?
$endgroup$
– 20gobbledigook08
Dec 2 '18 at 13:41




$begingroup$
@amd Ah, I see. Then for any polynomial with degree of two, I could have a quadratic form then?
$endgroup$
– 20gobbledigook08
Dec 2 '18 at 13:41












$begingroup$
@reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though!
$endgroup$
– 20gobbledigook08
Dec 2 '18 at 13:50




$begingroup$
@reuns Could you please explain what $A_{ij}$ suppose to mean? Hehe. Thanks for the help though!
$endgroup$
– 20gobbledigook08
Dec 2 '18 at 13:50












$begingroup$
en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms
$endgroup$
– reuns
Dec 2 '18 at 22:16




$begingroup$
en.wikipedia.org/wiki/Quadratic_form#Real_quadratic_forms
$endgroup$
– reuns
Dec 2 '18 at 22:16










1 Answer
1






active

oldest

votes


















0












$begingroup$

A function $F:EtoBbb R $ is a quadratic form if
$$G (x,y)=frac 12 (F (x+y)-F (x)-F (y)) $$
is bilinear.
In that case $G $ is a symmetric bilinear form and $F (x)=G (x,x) $.
Moreover, $F$ is said to be non singular whenever $G (x,y)=0$ for all $y $ implies $x=0$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 15:58










  • $begingroup$
    See for example here.
    $endgroup$
    – Fabio Lucchini
    Dec 2 '18 at 17:51











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

A function $F:EtoBbb R $ is a quadratic form if
$$G (x,y)=frac 12 (F (x+y)-F (x)-F (y)) $$
is bilinear.
In that case $G $ is a symmetric bilinear form and $F (x)=G (x,x) $.
Moreover, $F$ is said to be non singular whenever $G (x,y)=0$ for all $y $ implies $x=0$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 15:58










  • $begingroup$
    See for example here.
    $endgroup$
    – Fabio Lucchini
    Dec 2 '18 at 17:51
















0












$begingroup$

A function $F:EtoBbb R $ is a quadratic form if
$$G (x,y)=frac 12 (F (x+y)-F (x)-F (y)) $$
is bilinear.
In that case $G $ is a symmetric bilinear form and $F (x)=G (x,x) $.
Moreover, $F$ is said to be non singular whenever $G (x,y)=0$ for all $y $ implies $x=0$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 15:58










  • $begingroup$
    See for example here.
    $endgroup$
    – Fabio Lucchini
    Dec 2 '18 at 17:51














0












0








0





$begingroup$

A function $F:EtoBbb R $ is a quadratic form if
$$G (x,y)=frac 12 (F (x+y)-F (x)-F (y)) $$
is bilinear.
In that case $G $ is a symmetric bilinear form and $F (x)=G (x,x) $.
Moreover, $F$ is said to be non singular whenever $G (x,y)=0$ for all $y $ implies $x=0$.






share|cite|improve this answer











$endgroup$



A function $F:EtoBbb R $ is a quadratic form if
$$G (x,y)=frac 12 (F (x+y)-F (x)-F (y)) $$
is bilinear.
In that case $G $ is a symmetric bilinear form and $F (x)=G (x,x) $.
Moreover, $F$ is said to be non singular whenever $G (x,y)=0$ for all $y $ implies $x=0$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 2 '18 at 12:57

























answered Dec 2 '18 at 12:52









Fabio LucchiniFabio Lucchini

7,83811426




7,83811426












  • $begingroup$
    Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 15:58










  • $begingroup$
    See for example here.
    $endgroup$
    – Fabio Lucchini
    Dec 2 '18 at 17:51


















  • $begingroup$
    Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
    $endgroup$
    – 20gobbledigook08
    Dec 2 '18 at 15:58










  • $begingroup$
    See for example here.
    $endgroup$
    – Fabio Lucchini
    Dec 2 '18 at 17:51
















$begingroup$
Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
$endgroup$
– 20gobbledigook08
Dec 2 '18 at 15:58




$begingroup$
Hello! Could I ask the reference to this definition of nonsingular? Thanks a bunch!
$endgroup$
– 20gobbledigook08
Dec 2 '18 at 15:58












$begingroup$
See for example here.
$endgroup$
– Fabio Lucchini
Dec 2 '18 at 17:51




$begingroup$
See for example here.
$endgroup$
– Fabio Lucchini
Dec 2 '18 at 17:51


















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