1.7.24 Linear Transformations

Condition: Vectors \ (\ Vec {p} \) and \ (\ vec {q} \) Euclidean space \ (e_ {4} \) with the coordinates in the basis \ (\ overrightarrow {a_ {1}}}}}}}}} \ overrightarrow {a_ {2}, \ overrightarrow {a_ {3}}, \ overrightarrow {a_ {4} \), the vectors of which are determined relative to some orthonomated basis of this space. 1) Applying the process of orthogonalization, orthonomate the basis \ (\ left \ {\ overrightarrow {a_}} \ right \} \) (obtained basis \ (the resulting basis -\ Left \ {\ overrightarrow {b_ {j}} \ right \} \)). 2) Find the matrix of the transition from the obtained orthonomated basis \ (\ left \ {\ overrightarrow {b _ \ jmath}} \ right \} \) to the initial 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 orthonomated basis. 4) Calculate the scalar work \ ((\ 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 \\