1. ## Linear algebra problem

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

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

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

3. 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.

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

I am not sure how to start this. I'm guessing I need to use induction but I don't know how.
Hint: $\displaystyle \sqrt{a+b} \leq \sqrt{a}+\sqrt{b}$ for $\displaystyle a,b\geq 0$.

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

6. Originally Posted by paulrb
Yes, my problem is proving that part. I didn't figure it out.
Let $\displaystyle a=x^2,b=y^2$ where $\displaystyle x,y>0$.
Therefore, you want to prove $\displaystyle \sqrt{x^2+y^2} \leq \sqrt{x^2} + \sqrt{y^2} = x + y$.

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

7. 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.

8. 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.