Linear algebra problem

**paulrb** · February 25th 2009, 01:30 PM

Let ${x}=[x_{1}, x_{2}, ... ,x_{n}]$ be a vector in $\mathbb{R}^{n}$ . Prove that $||x|| \leq\sum_{i=1}^n |x_{i}|$ .

I am not sure how to start this. I'm guessing I need to use induction but I don't know how.

**arbolis** · February 25th 2009, 01:34 PM

First I'd write down what is worth $||x||$ and then compare it with $\sum_{i=1}^n |x_{i}|$ .
Do you know what is worth $||x||$ ?

**paulrb** · February 25th 2009, 03:41 PM

Yes, it's the square root of x1^2+x2^2+...+xn^2.

But I don't understand how to compare the two...if I use induction I can show it's true for n = 1 but I don't know how to show it's true for n+1.

**ThePerfectHacker** · February 26th 2009, 09:56 PM

Originally Posted by paulrb

Let ${x}=[x_{1}, x_{2}, ... ,x_{n}]$ be a vector in $\mathbb{R}^{n}$ . Prove that $||x|| \leq\sum_{i=1}^n |x_{i}|$ .

I am not sure how to start this. I'm guessing I need to use induction but I don't know how.

Hint: $\sqrt{a+b} \leq \sqrt{a}+\sqrt{b}$ for $a,b\geq 0$ .

**paulrb** · February 27th 2009, 07:12 AM

Yes, my problem is proving that part. I didn't figure it out.

**ThePerfectHacker** · February 27th 2009, 04:02 PM

Originally Posted by paulrb

Yes, my problem is proving that part. I didn't figure it out.

Let $a=x^2,b=y^2$ where $x,y>0$ .
Therefore, you want to prove $\sqrt{x^2+y^2} \leq \sqrt{x^2} + \sqrt{y^2} = x + y$ .

We will prove this in a geometric way. Imagine a right triangle with legs $x,y$ . The third leg is $\sqrt{x^2+y^2}$ by Pythagorean theorem. However, by the triangle inequality it means the sum of any two sides must be great than the third sides, therefore, $\sqrt{x^2+y^2}\leq x+y$ . Can you prove your vector inequality now?

**paulrb** · February 27th 2009, 07:07 PM

Yes, that makes sense to me, and I was able to prove the rest. Thank you.

I am having difficulty with the proofs in this class :/ I have never had to do proofs before. I can do them for the homework by studying the book and getting help from others, but today I had a test and didn't do well. There were 4 proofs, and I was only able to do two and most of the third, before running out of time.

Do you know of a book or some other resource to help me get more practice? My book has some proofs (not many), yet my professor is almost entirely focused on proofs. I can understand when she writes them on the board but that is completely different from inventing my own.

**ThePerfectHacker** · February 27th 2009, 08:04 PM

Originally Posted by paulrb

There were 4 proofs, and I was only able to do two and most of the third, before running out of time.

Do you know of a book or some other resource to help me get more practice?

You usually get better at proofs the more you see them.

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