# 1.110: The Gram-Schmidt Procedure

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In this exercise the Gram-Schmidt method will be used to create an orthonormal basis set from the following vectors which are neither normalized nor orthogonal.

$\begin{matrix} u1 = \begin{pmatrix} 1 + i \\ 1 \\ i \end{pmatrix} & u2 = \begin{pmatrix} i \\ 3 \\ 1 \end{pmatrix} & u3 = \begin{pmatrix} 0 \\ 28 \\ 0 \end{pmatrix} \end{matrix} \nonumber$

Demonstrate that the vectors are not normalized and are not orthogonal.

$\begin{matrix} \left( \overline{u1} \right)^T u1 = 4 & \left( \overline{u2} \right)^T u2 = 11 & \left( \overline{u3} \right)^T u3 = 784 \\ \left( \overline{u1} \right)^T u2 = 4 & \left( \overline{u1} \right)^T u3 = 28 & \left( \overline{u2} \right)^T u3 = 84 \end{matrix} \nonumber$

Using the first vector make u2 orthogonal to it by subtracting its projection on u1.

$u2 = u2 - \frac{ \left( \overline{u1} \right)^T u2}{ \left( \overline{u1} \right)^T u1} u1 \nonumber$

Make u3 orthogonal to u1 and u2 by subtracting its projection on u1 and u2.

$u3 = u3 - \frac{ \left( \overline{u1} \right)^T u3}{ \left( \overline{u1} \right)^T u1} u1 - \frac{ \left( \overline{u2} \right)^T u3}{ \left( \overline{u2} \right)^T u2} u2 \nonumber$

Finally, normalize the new orthogonal vectors.

$\begin{matrix} u1 = \frac{u1}{ \sqrt{ \left( \overline{u1} \right)^T u1}} & u2 = \frac{u2}{ \sqrt{ \left( \overline{u2} \right)^T u2}} & u3 = \frac{u3}{ \sqrt{ \left( \overline{u3} \right)^T u3}} \end{matrix} \nonumber$

Demonstrate that an orthonormal basis set has been created.

$\begin{matrix} \left( \overline{u1} \right)^T u1 = 1 & \left( \overline{u2} \right)^T u2 = 1 & \left( \overline{u3} \right)^T u3 = 1 \\ \left( \overline{u1} \right)^T u2 = 0 & \left( \overline{u1} \right)^T u3 = 0 & \left( \overline{u2} \right)^T u3 = 0 \end{matrix} \nonumber$

Display the orthonormal basis set.

$\begin{matrix} u1 = \begin{pmatrix} 0.5 + 0.5i \\ 0.5 \\ 0.5i \end{pmatrix} & u2 = \begin{pmatrix} -0.378 \\ 0.756 \\ 0.378 - 0.378i \end{pmatrix} & u3 = \begin{pmatrix} 0.085 - 0.592i \\ 0.423 \\ -0.676 + 0.085i \end{pmatrix} \end{matrix} \nonumber$

1.110: The Gram-Schmidt Procedure is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.