Gram-Schmidt Process

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  1. Determine if the set of vectors is orthogonal and orthonormal. If a set is only orthogonal, normalize the vectors to produce an orthonormal set.   solution

    2. \[\left[\begin{array}{r}
    -2/3 \\ 1/3 \\ 2/3 \end{array}\right],\left[\begin{array}{r}
    1/3 \\
    2/3 \\
    0
    \end{array}\right]\]

  2. The given set is a basis for a subspace \(W .\) Use the Gram-Schmidt process to produce an orthogonal basis for \(W\).

    3. (a) \(\left[\begin{array}{l}0 \\ 4 \\ 2\end{array}\right],\left[\begin{array}{r}5 \\ 6 \\ -7\end{array}\right]\)  solution

    (b) \(\left[\begin{array}{r}-1 \\ 3 \\ 1 \\ 1\end{array}\right], \left[\begin{array}{r}6 \\ -8 \\ -2 \\ -4 \end{array}\right], \left[\begin{array}{r}6 \\ 3\\ 6 \\ -3 \end{array}\right] \)    solution

     

  3. Find an orthonormal basis of the subspace spanned by the vectors \(\left[\begin{array}{r}3 \\ 6 \\ 0\end{array}\right],\left[\begin{array}{c} 1 \\ 2 \\ 2 \end{array}\right]\).   solution

  4. Factor the following matrix \(A\) as \( A=QR \) where the columns of \( Q \) are orthonormal vectors found by the columns of \(A \).   \( A=\left[\begin{array}{rr} 3 & 1 \\ 6 & 2 \\ 0 & 2 \end{array}\right] \)    solution