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Math Help - Linear independence and orthogonality of vectors

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    Linear independence and orthogonality of vectors

    okay, so this is really two questions, the first is kind of simple:

    if a+b=0, a+c=0 and c=b can it be shown that a+b+c=0? (without the use of determinants)

    The second is: what is an example of a set of vectors that are linearly independent but not orthogonal? (graphical interpretation would be appreciated)
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    GJA
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    Re: Linear independence and orthogonality of vectors

    Hi guybrush92,

    Knowing a + b = 0, a+c=0 and b=c (which follows from the first two equations) it does not always follow that a+b+c=0. Try playing around with some specific values for a, b \& c to see why.

    For the second question, thinking about \mathbb{R}^{2} might be the easiest place to look. Geometrically, two vectors are linearly indepdent in \mathbb{R}^{2} if they don't lie on the same line through the origin. It may be simplest to take one of the vectors to be [1,0]. To find a second vector that is linearly independent from [1,0] and is not orthogonal to it means we should pick a vector that does not sit on the x-axis and is not perpendicular to the x-axis.

    Does this get things going in the right direction? Let me know if anything is unclear. Good luck!
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