prove that specific linear transformation is bounded

Question

this is the last question I was not able to answer to prepare for my final.

Consider the linear transformation $T: \Bbb R^3 \to \Bbb R^2$ defined, for $x = (x_1, x_2, x_3) ∈ \Bbb R^3$ , by

$$T(x) = (x_2-x_3 \,,\,x_1+2x_2)$$

where $\Bbb R^3$ and $\Bbb R^2$ are endowed with the normal Euclidean norms.

Show that $T$ is bounded by finding a positive real number $M$ such that

$$||T(x)||≤ M||x|| \;\;\text{for all}\;\; x ∈ \Bbb R^3$$

Justify your answer.

You should not copy questions from your text book without at least showing some effort in where you ran into trouble, or without showing what you tried. — sxd, Mar 18 '13 at 22:05
If you're preparing for a final I bet you can tell us what you tried, where did you get stuck, etc....? — DonAntonio, Mar 18 '13 at 22:08
@DonAntonio, I just inspected all of his questions, seems like all of them are like these... — sxd, Mar 18 '13 at 22:40

score 3 · Accepted Answer · answered Mar 18 '13 at 22:34

3

Sami's solution is pretty nice. I present an alternative solution (of course in no way better or more elegant, but for a novice it might look a bit more convenient).

Please note, that it suffices to find an upper bound for $\|T(x)\|$ for all $x$ with $\|x\|=1$, because for arbitrary $x$ we have

$$ \begin{align} \|T(x)\| \leq M \|x\| \quad\Longleftrightarrow\quad \|T\left(\frac{x}{\|x\|}\right)\| \leq M \end{align} $$

where $\frac{x}{\|x\|}$ has norm 1. Now assume $\|x\|=1$, then each coordinate's absolute value is bounded by 1 and so

$$ \begin{align} \|T(x)\|^2 &= (x_2-x_3)^2 + (x_1 + 2x_2)^2 \\ &= 5x_2^2 + x_3^2 + x_1^2 - 2x_2 x_3 + 4x_2x_1 \\ &\leq 5 + 1 + 1 + 2 + 4 = 13 \end{align} $$

It follows that $\sqrt{13}$ satisfies the desired property.

answered Mar 18 '13 at 22:34

thomas

515

I assumed that $\mathbb{R}^3$ and $\mathbb{R}^2$ are endowed with the Euclidean norm. At least, that's what the OP wrote - or what are you referring to? – thomas Mar 18 '13 at 22:52
You're right, of course. Thanks. +1 – DonAntonio Mar 18 '13 at 22:53
Should it not be 5+1+1 - 2+4=9? – Omar Mar 19 '13 at 19:28
1

No, because we have $|x_2|<1$ and $|x_3|<1$, hence $|x_2x_3|<1$ and especially $-x_2x_3<1$, but NOT $-x_2x_3<-1$ – thomas Mar 19 '13 at 22:34

score 2 · Answer 2 · 2013-03-18T22:26:48.203

2

The norm $1$ is defined by $$||(x,y,z)||_1=|x|+|y|+|z|.$$ and by Cauchy-Shwarz inequality we have $$||(x,y,z)||_1\leq\sqrt{3}||(x,y,z)||.$$ Now we have $$||T(x)||^2=(x_2-x_3)^2+(x_1+2x_2)^2\leq ||x||_1^2+||2x||_1^2=5||x||_1^2\leq 15||x||^2$$ hence $$||T(x)||\leq \sqrt{15}||x||.$$

edited Mar 18 '13 at 22:26

answered Mar 18 '13 at 22:19

1

Why using the $1$ norm? Also, it should be $\leq \sqrt{3}$ by CS. – Julien Mar 18 '13 at 22:27
@julien I used the $1$ norm just for the sake, in fact, I've only used the notation that it seems lighter. – Mar 19 '13 at 05:50

prove that specific linear transformation is bounded

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