I was reading about projection mapping in linear algebra between finite dimensional vector spaces. There were given a condition that my space should be written as direct sum of two of its subspaces V and W to define projection onto V along W. Now my question is
we can take projections of x=y plane in yz plane in 3D euclidean geometry, this case after x=y would be precisely yz plane.
But as per our definition of projection mapping, R^3 should be writen as direct sum of x=y plane and xz plane, but here is not so.
Are they means different projections? I mean in euclidean 3D geometry we can take projections of plane onto a plane, but here in the definition of linear algebra about projection we cant take like this in 3D.
So are not they denotes same projection?
May be my concepts are not clear at all. Please help me to clear out. Thanks in advance.