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Define a new addition and multiplication and check the equality

0

$begingroup$

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

share|cite|improve this question

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 | 

0

$begingroup$

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

share|cite|improve this question

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

share|cite|improve this question

asked Mar 27 at 4:35

K JayK Jay

336

$endgroup$

share|cite|improve this question

asked Mar 27 at 4:35

K JayK Jay

336

share|cite|improve this question

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

1 Answer
1

active

oldest

votes

1

$begingroup$

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 $V$.

1. Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
   $$
   u = (x,y) quad textand quad v =(a,b).
   $$
   


2. Compute $u+v$ by using your newly defined addition.


3. Compute $k(u+v)$ by using your newly defined multiplication.


4. Compute $ku$ and $kv$ by using your newly defined multiplication.


5. Compute $ku + kv$ by using your newly defined addition.


6. 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.

share|cite|improve this answer

answered Mar 27 at 9:02

DiglettDiglett

1,1081521

$endgroup$

add a comment | 

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1

$begingroup$

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 $V$.

1. Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
   $$
   u = (x,y) quad textand quad v =(a,b).
   $$
   


2. Compute $u+v$ by using your newly defined addition.


3. Compute $k(u+v)$ by using your newly defined multiplication.


4. Compute $ku$ and $kv$ by using your newly defined multiplication.


5. Compute $ku + kv$ by using your newly defined addition.


6. 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.

share|cite|improve this answer

answered Mar 27 at 9:02

DiglettDiglett

1,1081521

$endgroup$

add a comment | 

1

$begingroup$

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 $V$.

1. Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
   $$
   u = (x,y) quad textand quad v =(a,b).
   $$
   


2. Compute $u+v$ by using your newly defined addition.


3. Compute $k(u+v)$ by using your newly defined multiplication.


4. Compute $ku$ and $kv$ by using your newly defined multiplication.


5. Compute $ku + kv$ by using your newly defined addition.


6. 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.

share|cite|improve this answer

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 $V$.

1. Since $u,v in V$, there are real numbers $a,b,x,y in mathbbR$ such that
   $$
   u = (x,y) quad textand quad v =(a,b).
   $$
   


2. Compute $u+v$ by using your newly defined addition.


3. Compute $k(u+v)$ by using your newly defined multiplication.


4. Compute $ku$ and $kv$ by using your newly defined multiplication.


5. Compute $ku + kv$ by using your newly defined addition.


6. 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.

share|cite|improve this answer

answered Mar 27 at 9:02

DiglettDiglett

1,1081521

$endgroup$

share|cite|improve this answer

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