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Linearly independent but not orthogonal, how come?

  1. Jul 16, 2008 #1
    Hi everyone, I was reading about Gram-Schmidt process of converting two linearly independent vectors into orthogonal basis. But, as I understand, if two vectors are linearly independent then they are orthogonal! isn't that right??
    Could anybody explain... please.
  2. jcsd
  3. Jul 16, 2008 #2
    No. What does it mean to have a set of linearly independent vectors? What does it mean to have a set of orthogonal vectors?
  4. Jul 16, 2008 #3
    As I understand, a set of linearly independent vectors means that it is not possible to write any of them in terms of the others.
    a set of orthogonal vectors means that the dot product of any two of them is zero. which in turn means that they are independent of each other, right? that's what confuses me.
  5. Jul 16, 2008 #4


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    Vectors which are orthogonal to each other are linearly independent. But this does not imply that all linearly independent vectors are also orthogonal. Take i+j for example. The linear span of that i+j is k(i+j) for all real values of k. and you can visualise it as the vector stretching along the x-y plane in a northeast and southwest direction. However, there does not exist any value of k such that k(i+j) = i. i and i+j are linearly independent, but not orthogonal.
  6. Jul 16, 2008 #5


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    For example, in R2, the vectors <1, 0> and <1, 1,> are independent since the only way to have a<1, 0>+ b<1, 1>= 0 is to have a= 0 and b= 0. But they are NOT "orthogonal"- the angle between them is 45 degrees, not 90.

    As Defennndeer said, if two vectors are orthogonal, then they are linearly independent but it does NOT work the other way.
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