1.7.24 Linear transformations

Condition: Given vectors \( \vec{p} \) and \( \vec{q} \) of the Euclidean space \( E_{4} \) with coordinates in the basis \( \overrightarrow{a_{1}}, \overrightarrow{a_{2}}, \overrightarrow{a_{3}}, \overrightarrow{a_{4}} \), whose vectors are defined with respect to some orthonormal basis of this space. 1) Applying the orthogonalization process, orthonormalize the basis \( \left\{\overrightarrow{a_{i}}\right\} \) (the resulting basis \( -\left\{\overrightarrow{b_{j}}\right\} \) ). 2) Find the transition matrix from the resulting orthonormal basis \( \left\{\overrightarrow{b_{\jmath}}\right\} \) to the original basis \( \left\{\overrightarrow{a_{l}}\right\},\left(T_{b_{j} \rightarrow a_{i}}\right) \). 3) Find the coordinates of the vectors \( \vec{p} \) and \( \vec{q} \quad \) in this orthonormal basis. 4) Calculate the scalar product \( (\vec{p}, \vec{q}) \). 5) Calculate the angle between the vectors \( \vec{p} \) and \( \vec{q} \). \[ \vec{p}=\left[\begin{array}{c} 7 \\ 1 \\ 2 \\ -2 \end{array}\right] ; \vec{q}=\left[\begin{array}{l} 4 \\ 2 \\ 2 \\