Define a new addition and multiplication and check the equality Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Proving Crazy Vector Space with addition and multiplicationMultiplication and Addition tables the following:Closed under vector addition and scalar multiplicationDo addition and multiplication define a structure of a field?Check if the equality holdsLess suggestive terms for “vector addition” and “scalar multiplication”A counterexample that shows addition and scalar multiplication is not enough for a vector space?Vector space with new scalar multiplication defined over finite field still vector space?non-zero Vector space with new scalar multiplication defined over finite field still vector space?how to check for closed addition and closed multiplication
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Define a new addition and multiplication and check the equality
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Proving Crazy Vector Space with addition and multiplicationMultiplication and Addition tables the following:Closed under vector addition and scalar multiplicationDo addition and multiplication define a structure of a field?Check if the equality holdsLess suggestive terms for “vector addition” and “scalar multiplication”A counterexample that shows addition and scalar multiplication is not enough for a vector space?Vector space with new scalar multiplication defined over finite field still vector space?non-zero Vector space with new scalar multiplication defined over finite field still vector space?how to check for closed addition and closed multiplication
$begingroup$
Consider the set $V$ = $(x,y): x,y in mathbbR$.
Define a new addition and multiplication on $V$ by $(x,y) + (a,b) = (x+a-2,y+b)$ and $k(x,y) = (kx-2k-2x+6,ky)$.
Check whether the following equality holds for all $u,v in V$ and all $k in mathbbR$:
$$k(u+v)=ku+kv$$
linear-algebra
asked Mar 27 at 4:35
K JayK Jay
336
$endgroup$
add a comment |
$begingroup$
Consider the set $V$ = $(x,y): x,y in mathbbR$.
Define a new addition and multiplication on $V$ by $(x,y) + (a,b) = (x+a-2,y+b)$ and $k(x,y) = (kx-2k-2x+6,ky)$.
Check whether the following equality holds for all $u,v in V$ and all $k in mathbbR$:
$$k(u+v)=ku+kv$$
linear-algebra
asked Mar 27 at 4:35
K JayK Jay
336
$endgroup$
$begingroup$
What is your question ?
$endgroup$
– Fred
Mar 27 at 6:20
$begingroup$
Well that's what I was trying to understand. I dont know what this question is really asking. But I think I am supposed to check the equality using the new addition and multiplication definitions provided.
$endgroup$
– K Jay
Mar 27 at 8:22
$begingroup$
@KJay: You're right, you need to check whether this equality is true or not. Was that your question or do you not know how to proceed?
$endgroup$
– Diglett
Mar 27 at 8:35
$begingroup$
Yes, I don't know where to start with this @Diglett
$endgroup$
– K Jay
Mar 27 at 8:42
add a comment |
$begingroup$
Consider the set $V$ = $(x,y): x,y in mathbbR$.
Define a new addition and multiplication on $V$ by $(x,y) + (a,b) = (x+a-2,y+b)$ and $k(x,y) = (kx-2k-2x+6,ky)$.
Check whether the following equality holds for all $u,v in V$ and all $k in mathbbR$:
$$k(u+v)=ku+kv$$
linear-algebra
asked Mar 27 at 4:35
K JayK Jay
336
$endgroup$
Consider the set $V$ = $(x,y): x,y in mathbbR$.
Define a new addition and multiplication on $V$ by $(x,y) + (a,b) = (x+a-2,y+b)$ and $k(x,y) = (kx-2k-2x+6,ky)$.
Check whether the following equality holds for all $u,v in V$ and all $k in mathbbR$:
$$k(u+v)=ku+kv$$
linear-algebra
linear-algebra
asked Mar 27 at 4:35
K JayK Jay
336
asked Mar 27 at 4:35
K JayK Jay
336
asked Mar 27 at 4:35
K JayK Jay
336
asked Mar 27 at 4:35
K JayK Jay
336
asked Mar 27 at 4:35
K JayK Jay
336
336
$begingroup$
What is your question ?
$endgroup$
– Fred
Mar 27 at 6:20
$begingroup$
Well that's what I was trying to understand. I dont know what this question is really asking. But I think I am supposed to check the equality using the new addition and multiplication definitions provided.
$endgroup$
– K Jay
Mar 27 at 8:22
$begingroup$
@KJay: You're right, you need to check whether this equality is true or not. Was that your question or do you not know how to proceed?
$endgroup$
– Diglett
Mar 27 at 8:35
$begingroup$
Yes, I don't know where to start with this @Diglett
$endgroup$
– K Jay
Mar 27 at 8:42
add a comment |
$begingroup$
What is your question ?
$endgroup$
– Fred
Mar 27 at 6:20
$begingroup$
Well that's what I was trying to understand. I dont know what this question is really asking. But I think I am supposed to check the equality using the new addition and multiplication definitions provided.
$endgroup$
– K Jay
Mar 27 at 8:22
$begingroup$
@KJay: You're right, you need to check whether this equality is true or not. Was that your question or do you not know how to proceed?
$endgroup$
– Diglett
Mar 27 at 8:35
$begingroup$
Yes, I don't know where to start with this @Diglett
$endgroup$
– K Jay
Mar 27 at 8:42
$begingroup$
What is your question ?
$endgroup$
– Fred
Mar 27 at 6:20
$begingroup$
What is your question ?
$endgroup$
– Fred
Mar 27 at 6:20
$begingroup$
Well that's what I was trying to understand. I dont know what this question is really asking. But I think I am supposed to check the equality using the new addition and multiplication definitions provided.
$endgroup$
– K Jay
Mar 27 at 8:22
$begingroup$
Well that's what I was trying to understand. I dont know what this question is really asking. But I think I am supposed to check the equality using the new addition and multiplication definitions provided.
$endgroup$
– K Jay
Mar 27 at 8:22
$begingroup$
@KJay: You're right, you need to check whether this equality is true or not. Was that your question or do you not know how to proceed?
$endgroup$
– Diglett
Mar 27 at 8:35
$begingroup$
@KJay: You're right, you need to check whether this equality is true or not. Was that your question or do you not know how to proceed?
$endgroup$
– Diglett
Mar 27 at 8:35
$begingroup$
Yes, I don't know where to start with this @Diglett
$endgroup$
– K Jay
Mar 27 at 8:42
$begingroup$
Yes, I don't know where to start with this @Diglett
$endgroup$
– K Jay
Mar 27 at 8:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
We would like to check if the equation
$$
k(u+v) = ku + kv
$$
is true for all $u,v in V$ and all $k in mathbbR$.
Let me give you a blueprint for the solution of this problem:
Let $k$ be an arbitrary real number and $u,v$ be arbitrary elements in $V$.
- Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
$$
u = (x,y) quad textand quad v =(a,b).
$$ - Compute $u+v$ by using your newly defined addition.
- Compute $k(u+v)$ by using your newly defined multiplication.
- Compute $ku$ and $kv$ by using your newly defined multiplication.
- Compute $ku + kv$ by using your newly defined addition.
- Compare your results from 3. and 5. by comparing the entries of the vectors. This shows whether your equality is true or not.
- If the equality seems to be not true, try to find explicit $a,b,x,y$, so that the equality does not hold.
If you need further help, then you can let me know and I try to give you more hints.
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We would like to check if the equation
$$
k(u+v) = ku + kv
$$
is true for all $u,v in V$ and all $k in mathbbR$.
Let me give you a blueprint for the solution of this problem:
Let $k$ be an arbitrary real number and $u,v$ be arbitrary elements in $V$.
- Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
$$
u = (x,y) quad textand quad v =(a,b).
$$ - Compute $u+v$ by using your newly defined addition.
- Compute $k(u+v)$ by using your newly defined multiplication.
- Compute $ku$ and $kv$ by using your newly defined multiplication.
- Compute $ku + kv$ by using your newly defined addition.
- Compare your results from 3. and 5. by comparing the entries of the vectors. This shows whether your equality is true or not.
- If the equality seems to be not true, try to find explicit $a,b,x,y$, so that the equality does not hold.
If you need further help, then you can let me know and I try to give you more hints.
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
$endgroup$
add a comment |
$begingroup$
We would like to check if the equation
$$
k(u+v) = ku + kv
$$
is true for all $u,v in V$ and all $k in mathbbR$.
Let me give you a blueprint for the solution of this problem:
Let $k$ be an arbitrary real number and $u,v$ be arbitrary elements in $V$.
- Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
$$
u = (x,y) quad textand quad v =(a,b).
$$ - Compute $u+v$ by using your newly defined addition.
- Compute $k(u+v)$ by using your newly defined multiplication.
- Compute $ku$ and $kv$ by using your newly defined multiplication.
- Compute $ku + kv$ by using your newly defined addition.
- Compare your results from 3. and 5. by comparing the entries of the vectors. This shows whether your equality is true or not.
- If the equality seems to be not true, try to find explicit $a,b,x,y$, so that the equality does not hold.
If you need further help, then you can let me know and I try to give you more hints.
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
$endgroup$
add a comment |
$begingroup$
We would like to check if the equation
$$
k(u+v) = ku + kv
$$
is true for all $u,v in V$ and all $k in mathbbR$.
Let me give you a blueprint for the solution of this problem:
Let $k$ be an arbitrary real number and $u,v$ be arbitrary elements in $V$.
- Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
$$
u = (x,y) quad textand quad v =(a,b).
$$ - Compute $u+v$ by using your newly defined addition.
- Compute $k(u+v)$ by using your newly defined multiplication.
- Compute $ku$ and $kv$ by using your newly defined multiplication.
- Compute $ku + kv$ by using your newly defined addition.
- Compare your results from 3. and 5. by comparing the entries of the vectors. This shows whether your equality is true or not.
- If the equality seems to be not true, try to find explicit $a,b,x,y$, so that the equality does not hold.
If you need further help, then you can let me know and I try to give you more hints.
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
$endgroup$
We would like to check if the equation
$$
k(u+v) = ku + kv
$$
is true for all $u,v in V$ and all $k in mathbbR$.
Let me give you a blueprint for the solution of this problem:
Let $k$ be an arbitrary real number and $u,v$ be arbitrary elements in $V$.
- Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
$$
u = (x,y) quad textand quad v =(a,b).
$$ - Compute $u+v$ by using your newly defined addition.
- Compute $k(u+v)$ by using your newly defined multiplication.
- Compute $ku$ and $kv$ by using your newly defined multiplication.
- Compute $ku + kv$ by using your newly defined addition.
- Compare your results from 3. and 5. by comparing the entries of the vectors. This shows whether your equality is true or not.
- If the equality seems to be not true, try to find explicit $a,b,x,y$, so that the equality does not hold.
If you need further help, then you can let me know and I try to give you more hints.
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
answered Mar 27 at 9:02
DiglettDiglett
1,1081521
1,1081521
add a comment |
add a comment |
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$begingroup$
What is your question ?
$endgroup$
– Fred
Mar 27 at 6:20
$begingroup$
Well that's what I was trying to understand. I dont know what this question is really asking. But I think I am supposed to check the equality using the new addition and multiplication definitions provided.
$endgroup$
– K Jay
Mar 27 at 8:22
$begingroup$
@KJay: You're right, you need to check whether this equality is true or not. Was that your question or do you not know how to proceed?
$endgroup$
– Diglett
Mar 27 at 8:35
$begingroup$
Yes, I don't know where to start with this @Diglett
$endgroup$
– K Jay
Mar 27 at 8:42