I am stuck on the following problem:

Consider $\Bbb R^3$ with the standard inner product. Let $M$ be the subspace of $\Bbb R^3$ spanned by $(1,0,-1).$ Which of the following is a basis for the orthogonal complement of $M\,\,?$

  1. $\{(2,1,2),(4,2,4)\}$

  2. $\{(2,-1,2),(1,3,1),(-1,-1,-1)\}$

  3. $\{(1,0,1),(0,1,0)\}$

  4. $\{(1,2,1),(0,1,1)\}$

Can someone explain it?

  • $\begingroup$ very similar to this $\endgroup$
    – Mikasa
    Jun 28, 2013 at 15:47

1 Answer 1


You know that the complement has dimension 2. You also know that each vector in the complement is orthogonal to $(1,0,-1)$, so the basis vectors are too. In other words, you need to check

a) Which set of vectors are linearly independent and span a two dimensional subspace.

b) Which set consists of vectors orthogonal to $(1,0,-1)$.

  • $\begingroup$ then option 3 is the winner.Am i right? $\endgroup$
    – learner
    Jun 28, 2013 at 15:50
  • $\begingroup$ Yes, you are right. $\endgroup$ Jun 28, 2013 at 16:27

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