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Suppose that $v \in \mathbb{R}^n$ is a vector orthogonal to all vectors $x \in \mathbb{R}^n$. Prove that $v = 0$.

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  • $\begingroup$ Might I ask what curvy E is? Thanks. $\endgroup$
    – awllower
    Commented Apr 7, 2013 at 7:20
  • $\begingroup$ Take inner products. You can do this by looking at a general vector in $\mathbb{R}^n$ or by considering $<v,v>$. $\endgroup$
    – user27182
    Commented Apr 7, 2013 at 11:04

4 Answers 4

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Hint: What is |$v$ $\bullet$ $v$| ?

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if v is orthogonal to all of vector of R then for each standarad base we must have $v_i=v*e_i=0$ so v=0

sign * means inner product

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If $v$ is orthogonal to all vectors in $\mathbb{R}^n$, then in particular it is orthogonal to itself. What does this, then, imply?

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If $v$ is orthogonal to all vectors in $\mathbb R^n$, then as mentioned above $v$ is self orthogonal but then $<v,v>=0$, Now you need to know the definition of inner product (or scalar product in this case).$<v,v>=0$ $\implies v=0$

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