Norm Verification for Vectors in R^2

[bugatti79]

What is the first step that you're not 100% sure about? Do you know the definition of x+y? If yes, then why not use it? Then use the definition of $\|\ \|_{\#}$. After that you're almost done.

I don't know how to write the definition of x+y apart from what I gave originally..thanks for your patience.

 

[Mark44]

4) $\forall x,y \in \mathbb R^2$ and $x_1, y_1 \in \mathbb R$
we have $x+y = x_1+2x_2 +y_1 +2y_2$ ...?

So, x and y are elements of R², meaning that you can write them as
x = <x₁, x₂>
y = <y₁, y₂>

Then what is x + y? It is NOT as you wrote above.

I don't know how to write the definition of x+y apart from what I gave originally..thanks for your patience.

 

[Fredrik]

I don't know how to write the definition of x+y apart from what I gave originally..thanks for your patience.

I don't recall seeing a definition of x+y in this thread.

Edit: If you meant the equality $x+y = x_1+2x_2 +y_1 +2y_2$ (which is extremely incorrect, since the left-hand side is a member of a different set than the right-hand side), then no, this looks nothing at all like the definition of x+y. It looks like it was inspired by the definition of $\|\ \|_{\#}$, but I see nothing in it that reminds me of the definition of x+y.

I'm going to bed early tonight, but I'm sure Mark44 or someone else can answer your next post.

 

[bugatti79]

I don't recall seeing a definition of x+y in this thread.

Edit: If you meant the equality $x+y = x_1+2x_2 +y_1 +2y_2$ (which is extremely incorrect, since the left-hand side is a member of a different set than the right-hand side), then no, this looks nothing at all like the definition of x+y. It looks like it was inspired by the definition of $\|\ \|_{\#}$, but I see nothing in it that reminds me of the definition of x+y.

I'm going to bed early tonight, but I'm sure Mark44 or someone else can answer your next post.

Yes, you are correct. Ok, thanks.

It is $\|x+y\|_\#=...$ that I should be focusing on...

 

[Fredrik]

Yes, but before you focus on $\|x+y\|_{\#}$, you should focus on x+y. Do you understand what the notation $\mathbb R^2$ means? What sort of objects are members of $\mathbb R^2$. How do you add one of those objects to another? Do you at least understand that the notation x+y is completely meaningless until it has been defined? What is the standard definition of x+y for $x,y\in\mathbb R^2$?

 

[Mark44]

To elaborate on what Fredrik said, how is addition usually defined for vectors in R²? That's basically what he's asking about for the expression x + y. This has nothing to do with norms.

 

[bugatti79]

To elaborate on what Fredrik said, how is addition usually defined for vectors in R²? That's basically what he's asking about for the expression x + y. This has nothing to do with norms.

Yes, but before you focus on $\|x+y\|_{\#}$, you should focus on x+y. Do you understand what the notation $\mathbb R^2$ means? What sort of objects are members of $\mathbb R^2$. How do you add one of those objects to another? Do you at least understand that the notation x+y is completely meaningless until it has been defined? What is the standard definition of x+y for $x,y\in\mathbb R^2$?

$\|x+y\| \le \|x_1+y_1\|+ \|2x_2 +2y_2\|$

$\le \|x_1+y_1\|+2\|x_2+y_2\|$

If the value of y is 0, isn't it still possible that the inequality holds? ie, the LHS can still be less than the RHS ie

$\|x\| \le \|x_1\|+ \|2x_2 \|$

So I am saying that axiom 4 holds. Is this how one examines it? Thanks

 

[micromass]

$\|x+y\| \le \|x_1+y_1\|+ \|2x_2 +2y_2\|$

$\le \|x_1+y_1\|+2\|x_2+y_2\|$

If the value of y is 0, isn't it still possible that the inequality holds? ie, the LHS can still be less than the RHS ie

$\|x\| \le \|x_1\|+ \|2x_2 \|$

So I am saying that axiom 4 holds. Is this how one examines it? Thanks

No. What you wrote makes no sense.

We asked you several times, how is x+y defined??

Let x=(x_1,y_1) and y=(y_1,y_2). Then how did we define

(x_1,x_2)+(y_1,y_2)

?

 

[Fredrik]

$\|x+y\| \le \|x_1+y_1\|+ \|2x_2 +2y_2\|$

$\le \|x_1+y_1\|+2\|x_2+y_2\|$

Try doing just one thing at a time. For all $x,y\in\mathbb R^2$, we have
$$\|x+y\|_{\#}=...$$ The first step is to use the definition of + to rewrite x+y as an ordered pair of real numbers. The second step is to use the definition of $\|\ \|_{\#}$. Then there's a third step. Don't try to do them all at once. Do them one at a time.

You also need to think about when to use the notation |something| and when to use the notation $\|\text{something}\|_{\#}$. Also, in each step, use an equality sign if what you have on the left is the same thing as what you have on the right. Use ≤ only if you have reason to think that the thing on the right may be greater than the thing on the left.

 

[Mark44]

In addition, in this problem ||x + y|| is completely irrelevant. The norm you appear to be working with is ||x + y||_#. Use the definition of this norm to first say what this expression equals, and then work on showing that the triangle inequality holds.

 

[bugatti79]

In addition, in this problem ||x + y|| is completely irrelevant. The norm you appear to be working with is ||x + y||_#. Use the definition of this norm to first say what this expression equals, and then work on showing that the triangle inequality holds.

Try doing just one thing at a time. For all $x,y\in\mathbb R^2$, we have
$$\|x+y\|_{\#}=...$$ The first step is to use the definition of + to rewrite x+y as an ordered pair of real numbers. The second step is to use the definition of $\|\ \|_{\#}$. Then there's a third step. Don't try to do them all at once. Do them one at a time.

You also need to think about when to use the notation |something| and when to use the notation $\|\text{something}\|_{\#}$. Also, in each step, use an equality sign if what you have on the left is the same thing as what you have on the right. Use ≤ only if you have reason to think that the thing on the right may be greater than the thing on the left.

$x+y=(x_1+y_1,2x_2+2y_2)$?

 

[Mark44]

$x+y=(x_1+y_1,2x_2+2y_2)$?

NO! Do you not understand how to add two vectors together? You should not be attempting to work problems about norms if you don't understand the basics of vector operations.

 

[Fredrik]

$x+y=(x_1+y_1,2x_2+2y_2)$?

No, this is wrong. Note that we're just doing addition here. Those functions that may or may not be norms have nothing to do with it.

 

[micromass]

Bugatti, I think we could help you better if you would explain us your situation. What course are you taking now? What courses did you already take??
I'm asking this because I feel you miss some preliminary knowledge. We can help you rectify it by suggesting things you should look at.

 

[bugatti79]

NO! Do you not understand how to add two vectors together? You should not be attempting to work problems about norms if you don't understand the basics of vector operations.

If v=(v1,v2) and w=(w1,w2) then v+w=(v1+w1,v2+w2)? Thats all I have done above?

No, this is wrong. Note that we're just doing addition here. Those functions that may or may not be norms have nothing to do with it.

Bugatti, I think we could help you better if you would explain us your situation. What course are you taking now? What courses did you already take??
I'm asking this because I feel you miss some preliminary knowledge. We can help you rectify it by suggesting things you should look at.

I am studying topics in analysis part time so I may be over tired from work. I have done PDE's and Calculus 1,2 last year.

 

[Mark44]

If v=(v1,v2) and w=(w1,w2) then v+w=(v1+w1,v2+w2)? Thats all I have done above?

No, it isn't. You were mixing vector addition with the || ||_# norm, producing a meaningless mush. Here is what I'm referring to.

x+y=x₁ +2x₂ +y₁ +2y₂

 

[bugatti79]

No, it isn't. You were mixing vector addition with the || ||_# norm, producing a meaningless mush. Here is what I'm referring to.

So for x,y in R^2 where x=(x1,x2) and y=(y1,y2)

$x+ y= <x_1+ 2y_1, x_2+ 2y_2>$?

 

[micromass]

So for x,y in R^2 where x=(x1,x2) and y=(y1,y2)

$x+ y= <x_1+ 2y_1, x_2+ 2y_2>$?

No! You wrote it correctly in in your post 45. Why do you keep insisting on the 2?

 

[bugatti79]

No! You wrote it correctly in in your post 45. Why do you keep insisting on the 2?

$x+y=(x_1+y_1, x_2+y_2)$

$\| x+y\|_{\#}= \| |x1|+|y1|, 2|x2|+2|y2| \|_{\#}$...?

 

[Fredrik]

$x+y=(x_1+y_1, x_2+y_2)$

$\| x+y\|_{\#}= \| |x1|+|y1|, 2|x2|+2|y2| \|_{\#}$...?

Here you're adding x and y correctly, but why don't you just insert the result you've found for x+y into $\|x+y\|_{\#}$, and then use the definition of $\|\ \|_{\#}$? What you wrote doesn't make any sense. The function $\|\ \|_{\#}$ takes a member of $\mathbb R^2$ as input. Here you used a real number as input. That doesn't make sense.

Edit: Why is there a comma in there? Did you mean $\|(|x_1|+|y_1|,2|x_2|+2|y_2|)\|_{\#}$? That doesn't make much sense either. I mean, it it's not a nonsense expression, but I have no idea why you would consider an ordered pair with those components.

 

[Mark44]

$x+y=(x_1+y_1, x_2+y_2)$

$\| x+y\|_{\#}= \| |x1|+|y1|, 2|x2|+2|y2| \|_{\#}$...?

Did you use the definition?

(i) || ||_#: R^2 defined by ||x||_#=|x_1|+2|x_2|

Doesn't look like it to me.

 

[bugatti79]

Did you use the definition?


Doesn't look like it to me.

Here you're adding x and y correctly, but why don't you just insert the result you've found for x+y into $\|x+y\|_{\#}$, and then use the definition of $\|\ \|_{\#}$? What you wrote doesn't make any sense. The function $\|\ \|_{\#}$ takes a member of $\mathbb R^2$ as input. Here you used a real number as input. That doesn't make sense.

Edit: Why is there a comma in there? Did you mean $\|(|x_1|+|y_1|,2|x_2|+2|y_2|)\|_{\#}$? That doesn't make much sense either. I mean, it it's not a nonsense expression, but I have no idea why you would consider an ordered pair with those components.

Now I am really confused. I know this is a big ask but is it not possible for one of you to write the answer and then I will query why you did it that way?. I have spent 3.5 hours on this trivial problem.

 

[bugatti79]

The function $\|\ \|_{\#}$ takes a member of $\mathbb R^2$ as input. Here you used a real number as input. That doesn't make sense.

OK, should it be

$\| x+y\|_{\#}= \| \|x\|+\|y\| \|_{\#}$

 

[Mark44]

That's against forum rules.

Here's an example, though.
Suppose x = <3, 2> and y = <-1, 5>
||x + y||_# = ||<2, 7>||_# = |2| + 2|7| = 2 + 14 = 16

I can justify every step above. Can you provide a reason for each step?

See if you can use this as a template to calculate a more general result: ||x + y||_#, with x = <x₁, x₂> and y = <y₁, y₂>

 

[Mark44]

OK, should it be

$\| x+y\|_{\#}= \| \|x\|+\|y\| \|_{\#}$

Way off.

 

[bugatti79]

That's against forum rules.

Here's an example, though.
Suppose x = <3, 2> and y = <-1, 5>
||x + y||_# = ||<2, 7>||_# = |2| + 2|7| = 2 + 14 = 16

I can justify every step above. Can you provide a reason for each step?

See if you can use this as a template to calculate a more general result: ||x + y||_#, with x = <x₁, x₂> and y = <y₁, y₂>

x=(x1,x2), y=(y1,y2) both in R^2

line 1 ||x+y||_#=||(x1+y1),(x2+y2)||_# then using definition of the norm || ||_# we have
= |x1+y1|+2|x2+y2|
=|x1|+|y1|+2|x2| +2|y2|
=|x1|+2|x2|+|y1|+2|y2|
= ||x||_#+||y||_#

I am not sure how to justify going from line 2 to line 3...?

At what stage do I play with the inequalities?

 

[micromass]

x=(x1,x2), y=(y1,y2) both in R^2

line 1 ||x+y||_#=||(x1+y1),(x2+y2)||_# then using definition of the norm || ||_# we have
= |x1+y1|+2|x2+y2|
=|x1|+|y1|+2|x2| +2|y2|
=|x1|+2|x2|+|y1|+2|y2|
= ||x||_#+||y||_#

I am not sure how to justify going from line 2 to line 3...?

At what stage do I play with the inequalities?

Line 2 to 3 is wrong. You won't have an [STRIKE]inequality[/STRIKE] equality. But perhaps you can use

|a+b|\leq |a|+|b|

for a,b\in \mathbb{R}.

 

[bugatti79]

That's against forum rules.

Here's an example, though.
Suppose x = <3, 2> and y = <-1, 5>
||x + y||_# = ||<2, 7>||_# = |2| + 2|7| = 2 + 14 = 16

I can justify every step above. Can you provide a reason for each step?

See if you can use this as a template to calculate a more general result: ||x + y||_#, with x = <x₁, x₂> and y = <y₁, y₂>

From ||x + y||_# = ||<2, 7>||_# because we have x+y=(x1+y1, x2+y2)

From ||<2, 7>||_# = |2| + 2|7| because we have the definition of the norm ||x ||_# as |x1 |+ 2 | x2|

 

[Mark44]

x=(x1,x2), y=(y1,y2) both in R^2

line 1 ||x+y||_#=||(x1+y1),(x2+y2)||_# then using definition of the norm || ||_# we have
= |x1+y1|+2|x2+y2|
=|x1|+|y1|+2|x2| +2|y2|
=|x1|+2|x2|+|y1|+2|y2|
= ||x||_#+||y||_#

I am not sure how to justify going from line 2 to line 3...?

As already noted by micromass, it's not justifiable.

Your first couple of steps were a good start, though, giving justifications for those steps. If you think about things from this perspective (giving a justification - definition, theorem, etc. - for each step) you will have made an important step toward thinking mathematically.

 

[Mark44]

From ||x + y||_# = ||<2, 7>||_# because we have x+y=(x1+y1, x2+y2)

From ||<2, 7>||_# = |2| + 2|7| because we have the definition of the norm ||x ||_# as |x1 |+ 2 | x2|

Yes, exactly right. This is precisely what we have been trying to get you to do.