Much help would be appreciated on these problems, thanks!
1.)
In
the orthogonal projections on the x-axis, y-axis and z-axis are
defined by
, and
respectively.
(a) Show that the orthogonal projections on the coordinate axes are linear
operators, and find their standard matrices.
(b) Show that if
→
is an orthogonal projection on one of the coordinate
axes, then for every vector x in
, the vector T(x) and x – T(x) are
orthogonal vectors.
(c) Make a sketch showing x and x - T(x) in the case where T is the orthogonal
projection on the x-axis.
2.) (a) Is a composition of one-to-one linear transformations one-to-one? Justify
your conclusion.
(b) Can the composition of a one-to-one linear transformation and a linear
transformation that is not one-to-one be one-to-one? Account for both possible
orders of composition and justify your conclusion.