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I have the following question, and I'd like to get some tips on how to write the proof. I know why it is, but I'm still not so great at writing it mathematically.

If $u$ is a vector in $\mathbb{R}^n$ that is orthogonal to every vector in $\mathbb{R}^n$, then $u$ must be the zero vector. Why?

I'm starting off like this, but I don't know if it's the right way to do it, or if it is and I just don't know how to continue.

\begin{align} (\exists u\in\mathbb{R}^n)(\forall u\in\mathbb{R}^n)[\text{u is orthogonal to v}]&\iff u\cdot v=0\\ &\iff ? \end{align}

From here, instinctively I want to divide both sides by $v$, but I don't know if there is such a thing as dividing a dot product.

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Let $e_i$ be the standard basis vectors of $\mathbb{R}^n$. What does the equation $u\cdot e_i=0$ tell us? –  Jared May 23 '13 at 15:48
    
Your title has the implication reversed from the body question. Please fix it or confirm and I will. –  Ross Millikan May 23 '13 at 15:50

3 Answers 3

up vote 10 down vote accepted

The dot product $\cdot$ is an unusual type of multiplication. It takes two vectors in and produces a scalar out. Imagine a mommy elephant and a daddy elephant giving birth to a giraffe.

There's no such thing as division in this context. You need to do the proof by looking at components of the vectors. $u=(u_1,u_2,\ldots, u_n), v=(v_1, v_2,\ldots, v_n)$, and $$u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n$$

Then, following Jared's hint, you can try specific values for $v$ and learn things about the components of $u$.

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LOL, I am going to use your ZOO terminology in my lin algebra class :) –  imranfat May 23 '13 at 15:51
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Could you explain to me the action of giraffes on elephants? :-) –  Sammy Black May 23 '13 at 16:13

Hint: what would it mean for a vector to be orthogonal to itself?

If the problem says "every other vector" as in the title (but not the quoted text) you could instead consider what it means for a given vector $\vec{u}$ to be orthogonal to multiples of itself, e.g. to $-\vec{u}$.

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Of course, for $-\vec u$ to be distinct from $\vec u$, we'd have to assume that $\vec u\ne\vec 0$. –  Cameron Buie May 23 '13 at 17:59
    
@Cameron True, but if $\vec{u} = \vec{0}$ then we're done. –  Trevor Wilson May 23 '13 at 18:02

Perhaps this is slightly off topic but it's worth explaining slightly more intuitively why you can't divide by a vector.

The key idea is to think about information (or more precisely numbers of constraints/equations). A dot product takes two vectors, both with $n$ numbers in them, and combines them in a particular way (related to projecting then onto each other) to give you one number. So $2n\to 1$.

Lots of information is clearly not being used in this function. There must be many different inputs giving the same output so it's hard to invert the process. But that's not unfamiliar, multiplication (or even addition) does something similar, taking $2\to 1$. Indeed $4\times 1=2\times 2$.

The idea of division normally works by saying "Okay, so suppose I know one of the inputs as well as the output. Can I figure out the other input?" The answer is yes. Schematically, you're now given two pieces of information at the end, so $2\to 2$ pieces of information. Then you have enough to figure out what you started with! (So long as you weren't just given a zero for the input, in the case of multiplication. It works perfectly for addition.)

But what about the dot product? If someone told you what one of the vectors was as well as the dot product, that's $n$ extra pieces of information. Therefore you now get $2n\to n+1$. But unless $n=1$ (normal multiplication) you still don't have enough data! The problem is everything in the vector you seek which is orthogonal to the input vector you were told is thrown away. You have a gap of $2n-(n+1)=n-1$ in your data. You have improved by just $1$ on your original ignorance.

But if you get another $k$ input vectors, and are told both them and their dot products with your mystery vector, you get $(k+1)n\to k(n+1)$. Clearly when $n=k$ you get a balance! Therefore, trying out dot products with $n$ different fixed vectors ought to be enough information (typically) to reverse the process.

So what is this magical new division? Well, you take $n$ vectors of $n$ entries, dot them all with the unknown vector, and collect the $n$ results. If you check it out, this is exactly matrix multiplication by an $n\times n$ matrix formed by taking rows of $A$ to be the vectors chosen. That is, you are given $A u=d$ where $d$ is the vector of dot products. So the division is actually just inverting the matrix $A$ to get $u=A^{-1} d$. Magic! And just as multiplication fails to be invertible when you times by zero, this fails if $A$ is not invertible; which is equivalent (as you might know) to the test input vectors all being linearly independent.

Since in this problem you know that all vectors dotted with $u$ give zero, you can choose any linearly independent set of test vectors and you'll find $u=A^{-1} \pmatrix{0 \\ 0 \\\vdots \\ 0}$ which vanishes. Hence $u$ is the zero vector.

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