Mixing
Product of functions in higher dimensions
$begingroup$
Suppose $f,g :mathbb R^ntomathbb R^m$ be two functions.
My question is that whether we can define product of $f$ and $g$ the i.e $f.g$.
My teacher says no. I wonder why not? We can define inner product or pointwise product of terms.
real-analysis calculus
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
asked Feb 1 at 19:53
IbrahimIbrahim
627
$endgroup$
add a comment |
$begingroup$
Suppose $f,g :mathbb R^ntomathbb R^m$ be two functions.
My question is that whether we can define product of $f$ and $g$ the i.e $f.g$.
My teacher says no. I wonder why not? We can define inner product or pointwise product of terms.
real-analysis calculus
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
asked Feb 1 at 19:53
IbrahimIbrahim
627
$endgroup$
add a comment |
$begingroup$
Suppose $f,g :mathbb R^ntomathbb R^m$ be two functions.
My question is that whether we can define product of $f$ and $g$ the i.e $f.g$.
My teacher says no. I wonder why not? We can define inner product or pointwise product of terms.
real-analysis calculus
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
asked Feb 1 at 19:53
IbrahimIbrahim
627
$endgroup$
Suppose $f,g :mathbb R^ntomathbb R^m$ be two functions.
My question is that whether we can define product of $f$ and $g$ the i.e $f.g$.
My teacher says no. I wonder why not? We can define inner product or pointwise product of terms.
real-analysis calculus
real-analysis calculus
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
asked Feb 1 at 19:53
IbrahimIbrahim
627
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
asked Feb 1 at 19:53
IbrahimIbrahim
627
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
edited Feb 1 at 19:55
Mundron Schmidt
7,4942729
7,4942729
asked Feb 1 at 19:53
IbrahimIbrahim
627
asked Feb 1 at 19:53
IbrahimIbrahim
627
asked Feb 1 at 19:53
IbrahimIbrahim
627
627
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The question is whether we can define a product of two elements in $mathbb R^m$.
The answer is: It depends, what you like to get. Naturally, you can define a lot of operations which combines two elements of $mathbb R^m$ like the dot product. You can even define operator from $mathbb R^mtimesmathbb R^mtomathbb R^m$. Further, you can do it such that $(mathbb R^m, +,cdot)$ is a ring like $mathbb Z$ by the componentwise multiplication.
But if you ask for a product $cdot:mathbb R^mtimesmathbb R^mtomathbb R^m$ such that $(mathbb R^m, +,cdot)$ forms a field then your teacher is right.
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
The question is whether we can define a product of two elements in $mathbb R^m$.
The answer is: It depends, what you like to get. Naturally, you can define a lot of operations which combines two elements of $mathbb R^m$ like the dot product. You can even define operator from $mathbb R^mtimesmathbb R^mtomathbb R^m$. Further, you can do it such that $(mathbb R^m, +,cdot)$ is a ring like $mathbb Z$ by the componentwise multiplication.
But if you ask for a product $cdot:mathbb R^mtimesmathbb R^mtomathbb R^m$ such that $(mathbb R^m, +,cdot)$ forms a field then your teacher is right.
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
$endgroup$
add a comment |
$begingroup$
The question is whether we can define a product of two elements in $mathbb R^m$.
The answer is: It depends, what you like to get. Naturally, you can define a lot of operations which combines two elements of $mathbb R^m$ like the dot product. You can even define operator from $mathbb R^mtimesmathbb R^mtomathbb R^m$. Further, you can do it such that $(mathbb R^m, +,cdot)$ is a ring like $mathbb Z$ by the componentwise multiplication.
But if you ask for a product $cdot:mathbb R^mtimesmathbb R^mtomathbb R^m$ such that $(mathbb R^m, +,cdot)$ forms a field then your teacher is right.
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
$endgroup$
add a comment |
$begingroup$
The question is whether we can define a product of two elements in $mathbb R^m$.
The answer is: It depends, what you like to get. Naturally, you can define a lot of operations which combines two elements of $mathbb R^m$ like the dot product. You can even define operator from $mathbb R^mtimesmathbb R^mtomathbb R^m$. Further, you can do it such that $(mathbb R^m, +,cdot)$ is a ring like $mathbb Z$ by the componentwise multiplication.
But if you ask for a product $cdot:mathbb R^mtimesmathbb R^mtomathbb R^m$ such that $(mathbb R^m, +,cdot)$ forms a field then your teacher is right.
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
$endgroup$
The question is whether we can define a product of two elements in $mathbb R^m$.
The answer is: It depends, what you like to get. Naturally, you can define a lot of operations which combines two elements of $mathbb R^m$ like the dot product. You can even define operator from $mathbb R^mtimesmathbb R^mtomathbb R^m$. Further, you can do it such that $(mathbb R^m, +,cdot)$ is a ring like $mathbb Z$ by the componentwise multiplication.
But if you ask for a product $cdot:mathbb R^mtimesmathbb R^mtomathbb R^m$ such that $(mathbb R^m, +,cdot)$ forms a field then your teacher is right.
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
answered Feb 1 at 20:02
Mundron SchmidtMundron Schmidt
7,4942729
7,4942729
add a comment |
add a comment |
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