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Math Help - Basis Orthogonal complement

  1. #1
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    Basis Orthogonal complement

    I need to find a basis for the orthogonal complement of the subspace \begin{array}{cc}t\\-t\\3t\end{array} that is orthogonal to the basis vector obtained by letting t = 1.

    Ok so i think i need to find a vector that is orthogonal to this one, which means that i just dot product with another vector. I just don't know what vector to use, or exactly what to do with the t bit.... Any help will be appreciated thank you in advance.
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  2. #2
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    Quote Originally Posted by Orbent View Post
    I need to find a basis for the orthogonal complement of the subspace \begin{array}{cc}t\\-t\\3t\end{array} that is orthogonal to the basis vector obtained by letting t = 1.

    Ok so i think i need to find a vector that is orthogonal to this one, which means that i just dot product with another vector. I just don't know what vector to use, or exactly what to do with the t bit.... Any help will be appreciated thank you in advance.

    I'm almost 100% sure I know what you meant, but college students must learn how to properly ask questions: what vector space are you talking about? What inner (or "dot") product is defined there? Why didn't you write your vector as \begin{pmatrix}t\\\!\!\!-t\\3t\end{pmatrix} and not like that weird-looking thing?...

    Tonio
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  3. #3
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    Well i didn't know how to put \begin{pmatrix}t\\\!\!\!-t\\3t\end{pmatrix} in. I am in engineering i learned about "dot product" before inner product so i associate Euclidean inner product with dot product. Since no vector space is mentioned in the question it is probably ok to assume it's euclidean. That being said i don't really know much about linear algebra, it is by far my weakest course. I hope i cleared everything up for you.
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  4. #4
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    Okay, "letting t= 1" in that gives \begin{pmatrix}1 \\ -1\\ 3\end{pmatrix}. A vector, \begin{pmatrix}a \\ b\\ c\end{pmatrix}, will be orthogonal to that if and only if \begin{pmatrix}a \\ b\\ c\end{pmatrix}\cdot\begin{pmatrix}1 \\ -1 \\ 3\end{pmatrix}= a- b+ 3c= 0

    From a- b+ 3c= 0, b= a+ 3c so \begin{pmatrix}a \\ b\\ c\end{pmatrix}= \begin{pmatrix}a \\ a+ 3c \\ c\end{pmatrix}= a\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}+ c\begin{pmatrix}0 \\ 3 \\ 1\end{pmatrix}.
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  5. #5
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    Thank you halls, i was heading along the right track, just couldn't think straight. Thank very much!
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