Question

3. Consider the set \{(4, -1, 1), (-1, 0, 4), (-4, -17, -1)\}. (a) Determine whether the vectors in the above set are orthogonal with respect to the Euclidean inner product on $\mathbb{R}^3$. (b) Show that the vectors form a basis for $\mathbb{R}^3$. (c) Apply the Gram-Schmidt orthonormalization process to transform the above basis for $\mathbb{R}^3$ into an orthonormal basis. (d) Express the vector $x = (1, -1, 0)$ as a linear combination of the orthonormal basis found in part (c). (Be warned that the coefficients in the linear combination may not necessarily be rational numbers.)

          3.
Consider the set \{(4, -1, 1), (-1, 0, 4), (-4, -17, -1)\}.
(a) Determine whether the vectors in the above set are orthogonal with
respect to the Euclidean inner product on $\mathbb{R}^3$.
(b) Show that the vectors form a basis for $\mathbb{R}^3$.
(c) Apply the Gram-Schmidt orthonormalization process to transform the
above basis for $\mathbb{R}^3$ into an orthonormal basis.
(d) Express the vector $x = (1, -1, 0)$ as a linear combination of the
orthonormal basis found in part (c).
(Be warned that the coefficients in the linear combination may not
necessarily be rational numbers.)

$3. Consider the set {(4, -1, 1), (-1, 0, 4), (-4, -17, -1)}. (a) Determine whether the vectors in the above set are orthogonal with respect to the Euclidean inner product on ℝ^3. (b) Show that the vectors form a basis for ℝ^3. (c) Apply the Gram-Schmidt orthonormalization process to transform the above basis for ℝ^3 into an orthonormal basis. (d) Express the vector x = (1, -1, 0) as a linear combination of the orthonormal basis found in part (c). (Be warned that the coefficients in the linear combination may not necessarily be rational numbers.)$

Added by Barbara D.

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