Why we need the same number of equations and variables?
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
2
I once read that to solve a system of equations, I need $n$ equations for $n$ variables.
I think I read it in a work by Euler. However, I have never managed to find a demonstration of this.
So, why is this true? What is the demonstration for any system of equations? I would appreciate a formal proof and another intuitive one, thanks in advance.
algebra-precalculus systems-of-equations
edited Sep 10 at 12:09
Harry Peter
5,52111439
asked Sep 8 at 12:48
Mattiu
745317
add a comment |Â
up vote
2
down vote
favorite
2
I once read that to solve a system of equations, I need $n$ equations for $n$ variables.
I think I read it in a work by Euler. However, I have never managed to find a demonstration of this.
So, why is this true? What is the demonstration for any system of equations? I would appreciate a formal proof and another intuitive one, thanks in advance.
algebra-precalculus systems-of-equations
edited Sep 10 at 12:09
Harry Peter
5,52111439
asked Sep 8 at 12:48
Mattiu
745317
3
The answer to your question for linear equations is the substance of the first part of most standard linear algebra courses. An answer here would just echo that material. You can start at the wikipedia page en.wikipedia.org/wiki/System_of_linear_equations
– Ethan Bolker
Sep 8 at 13:00
add a comment |Â
up vote
2
down vote
favorite
2
up vote
2
down vote
favorite
2
2
I once read that to solve a system of equations, I need $n$ equations for $n$ variables.
I think I read it in a work by Euler. However, I have never managed to find a demonstration of this.
So, why is this true? What is the demonstration for any system of equations? I would appreciate a formal proof and another intuitive one, thanks in advance.
algebra-precalculus systems-of-equations
edited Sep 10 at 12:09
Harry Peter
5,52111439
asked Sep 8 at 12:48
Mattiu
745317
I once read that to solve a system of equations, I need $n$ equations for $n$ variables.
I think I read it in a work by Euler. However, I have never managed to find a demonstration of this.
So, why is this true? What is the demonstration for any system of equations? I would appreciate a formal proof and another intuitive one, thanks in advance.
algebra-precalculus systems-of-equations
algebra-precalculus systems-of-equations
edited Sep 10 at 12:09
Harry Peter
5,52111439
asked Sep 8 at 12:48
Mattiu
745317
edited Sep 10 at 12:09
Harry Peter
5,52111439
asked Sep 8 at 12:48
Mattiu
745317
edited Sep 10 at 12:09
Harry Peter
5,52111439
edited Sep 10 at 12:09
Harry Peter
5,52111439
edited Sep 10 at 12:09
Harry Peter
5,52111439
5,52111439
asked Sep 8 at 12:48
Mattiu
745317
asked Sep 8 at 12:48
Mattiu
745317
asked Sep 8 at 12:48
Mattiu
745317
745317
3
The answer to your question for linear equations is the substance of the first part of most standard linear algebra courses. An answer here would just echo that material. You can start at the wikipedia page en.wikipedia.org/wiki/System_of_linear_equations
– Ethan Bolker
Sep 8 at 13:00
add a comment |Â
3
The answer to your question for linear equations is the substance of the first part of most standard linear algebra courses. An answer here would just echo that material. You can start at the wikipedia page en.wikipedia.org/wiki/System_of_linear_equations
– Ethan Bolker
Sep 8 at 13:00
3
3
The answer to your question for linear equations is the substance of the first part of most standard linear algebra courses. An answer here would just echo that material. You can start at the wikipedia page en.wikipedia.org/wiki/System_of_linear_equations
– Ethan Bolker
Sep 8 at 13:00
The answer to your question for linear equations is the substance of the first part of most standard linear algebra courses. An answer here would just echo that material. You can start at the wikipedia page en.wikipedia.org/wiki/System_of_linear_equations
– Ethan Bolker
Sep 8 at 13:00
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
It's not true in general.
Let $x_i in mathbbR$ where $i in 1, ldots, 100$.
$$sum_i=1^100x_i^2 =0.$$
We have $100$ variables and only one equation and we can solve it uniquely. That is $forall i in 1, ldots, 100, x_i=0$.
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
3
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
It's not true in general.
Let $x_i in mathbbR$ where $i in 1, ldots, 100$.
$$sum_i=1^100x_i^2 =0.$$
We have $100$ variables and only one equation and we can solve it uniquely. That is $forall i in 1, ldots, 100, x_i=0$.
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
3
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
add a comment |Â
up vote
2
down vote
It's not true in general.
Let $x_i in mathbbR$ where $i in 1, ldots, 100$.
$$sum_i=1^100x_i^2 =0.$$
We have $100$ variables and only one equation and we can solve it uniquely. That is $forall i in 1, ldots, 100, x_i=0$.
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
3
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
add a comment |Â
up vote
2
down vote
up vote
2
down vote
It's not true in general.
Let $x_i in mathbbR$ where $i in 1, ldots, 100$.
$$sum_i=1^100x_i^2 =0.$$
We have $100$ variables and only one equation and we can solve it uniquely. That is $forall i in 1, ldots, 100, x_i=0$.
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
It's not true in general.
Let $x_i in mathbbR$ where $i in 1, ldots, 100$.
$$sum_i=1^100x_i^2 =0.$$
We have $100$ variables and only one equation and we can solve it uniquely. That is $forall i in 1, ldots, 100, x_i=0$.
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
answered Sep 8 at 12:51
Siong Thye Goh
82.8k1456104
82.8k1456104
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
3
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
add a comment |Â
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
3
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
Well, the solution is unique in $mathbbR$, but not in $mathbbC$...
– Daniel Robert-Nicoud
Sep 8 at 12:53
3
3
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@DanielRobert-Nicoud Just change $x_i^2$ to $left|x_iright|$.
– user202729
Sep 8 at 12:57
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
@user202729 There's a deep difference between equations involving "nice" algebraic functions such as polynomials and those involving functions like taking the absolute value and similar functions. In the first case, the "$n$ equations for $n$ variables" can be formally understood in terms of algebraic geometry.
– Daniel Robert-Nicoud
Sep 8 at 14:01
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
Siong, can you explain me that, please?
– Mattiu
Sep 8 at 15:32
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
For the example that I constructed for real numbers, we know that for real number $x_i$, we have $x_i^2 ge 0$. If the sum of the squared terms is zero, then each term must be zero. I just constructed an example where we have one equation and multiple variables and yet it can be solved uniquely. In terms of linear algebra, if the $n$ equations are independent, then the coefficient matrix is invertible and we can invert a matrix.
– Siong Thye Goh
Sep 8 at 15:39
add a comment |Â
Â
draft saved
draft discarded
Â
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2909588%2fwhy-we-need-the-same-number-of-equations-and-variables%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
3
The answer to your question for linear equations is the substance of the first part of most standard linear algebra courses. An answer here would just echo that material. You can start at the wikipedia page en.wikipedia.org/wiki/System_of_linear_equations
– Ethan Bolker
Sep 8 at 13:00