Question

Find an orthonormal basis for the subspace of R³ spanned by

$\begin{Bmatrix} \begin{pmatrix} 1\\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1\\ 2 \\ 3 \end{pmatrix}\\ \\ \end{Bmatrix}$
Extend the basis you found to an orthonormal basis for R 3 (by adding a new vector or vectors). Is there a unique way to extend the basis you found to an orthonormal basis of R³ ? Explain.

Answer 1

"Let W be the subspace of IR^3 such that W = spam{(1, 1, 1), (1, 2, 3)} let U_1 = (1, 1, 1), U_2 = (1, 2, 3) therefore Using gram Schmidt process to obtain orthogonal bases so let V-1 = U_1 (1, 1, 1) V_2 = u_2 - < U_2, V_1 > /