Projection of a vector onto a subspace

[Grigorios Kostakos]

Let \(\bigl({\mathbb{C}^3}\,,\langle{-,-}\rangle\bigl)\) the inner product space with the usual inner product and the linear map \begin{align*} f:\,&\bigl({\mathbb{C}^3}\,,\langle{-,-}\rangle\bigl)\longrightarrow\bigl({\mathbb{C}^3}\,,\langle{-,-}\rangle\bigl)\,;\\ &\quad(v,w,z)\longmapsto\displaystyle{f(v,w,z)}=\bigl({(v+w)\,i\,,\,w+z\,,\,(v-z)\,i}\bigr)\,. \end{align*}

(1) Find the subspace \(\cal{V}\) which is generated by the vectors \(f(1+i,1-i,0)\), \(f(1,i, 1+i)\) and \(f(1-2i, 1-i, 3+3i)\).
Is this subspace equal with the image \({\rm{Im}}\,{f}\) of \(f\) ?

(2) Find the (orthogonal) projection of the vector \(\overrightarrow{\alpha}=(1,i,1+i)\) onto the subspace \(\cal{V}\).