Question

Find the (orthogonal) projection of \(\mathbf{x}:=(1,2,0)\) into the following subspaces:
a) The line spanned by \(\mathbf{u}:=(1,1,-1)\).
b) The plane spanned by \(\mathbf{u}:=(0,1,0)\) and \(\mathbf{v}:=(0,0,-2)\)
c) The plane spanned by \(\mathbf{u}:=(0,1,1)\) and \(\mathbf{v}:=(0,1,-2)\)
d) The plane spanned by \(\mathbf{u}:=(1,0,1)\) and \(\mathbf{v}:=(1,1,-1)\)
e) The plane spanned by \(\mathbf{u}:=(1,0,1)\) and \(\mathbf{v}:=(2,1,0)\).
f) The subspace spanned by \(\mathbf{u}:=(1,0,1), \mathbf{v}:=(2,1,0)\) and \(\mathbf{w}:=(1,1,0)\).