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Math Help - Linear algebra problem

  1. #1
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    Linear algebra problem

    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.
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  2. #2
    MHF Contributor arbolis's Avatar
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    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||?
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  3. #3
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    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.
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    Quote Originally Posted by paulrb View Post
    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.
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  5. #5
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    Yes, my problem is proving that part. I didn't figure it out.
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    Quote Originally Posted by paulrb View Post
    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?
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  7. #7
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    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.
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  8. #8
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    Quote Originally Posted by paulrb View Post
    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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