\[\mathbb{R}^3\]
, suppose we have a vector \[\mathbf{u}\]
.How do we construct the matrix that represents projection of any vector onto
\[\mathbf{u}\]
?We form the matrix whose columns are the vectors
\[ ( \frac{\begin{pmatrix}1\\0\\0\end{pmatrix} \cdot \begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}}{\sqrt{1^2+0^2+0^2} \sqrt{u^2_1+u^2_2+u^2_3}} ) \begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix} =\frac{1}{\sqrt{u^2_1+u^2_2+u^2_3}} \begin{pmatrix}u^2_1\\u_1u_2\\u_1u_3\end{pmatrix}\]
\[ ( \frac{\begin{pmatrix}0\\1\\0\end{pmatrix} \cdot \begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}}{\sqrt{1^2+0^2+0^2} \sqrt{u^2_1+u^2_2+u^2_3}} ) \begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix} =\frac{1}{\sqrt{u^2_1+u^2_2+u^2_3}} \begin{pmatrix}u_2u_1\\u^2_2\\u_2u_3\end{pmatrix}\]
\[ ( \frac{\begin{pmatrix}0\\0\\1\end{pmatrix} \cdot \begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}}{\sqrt{1^2+0^2+0^2} \sqrt{u^2_1+u^2_2+u^2_3}} ) \begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix} =\frac{1}{\sqrt{u^2_1+u^2_2+u^2_3}} \begin{pmatrix}u_3u_1\\u_3u_2\\u^2_3\end{pmatrix}\]
This gives the transformation matrix
\[\frac{1}{\sqrt{u^2_1+u^2_2+u^2_3}} \left| \begin{array}{ccc} u^2_1 & u_1 u_2 & u_1u_3 \\ u_2u_1 & u^2_2 &u_2u_3 \\ u_3u_1 & u_3u_2 &u^2_3 \end{array} \right| \]