Originally Posted by

**eddie25** Much help would be appreciated on these problems, thanks!

**1.)**

In $\displaystyle R^3$ the orthogonal projections on the x-axis, y-axis and z-axis are

defined by $\displaystyle T1(x, y, x) = (x, 0, 0), T2(x, y, z) = (0, y, 0)$, and $\displaystyle T3(x, y, z) = (0, 0, z)$

respectively.

**(a)** Show that the orthogonal projections on the coordinate axes are linear

operators, and find their standard matrices.

**(b)** Show that if $\displaystyle T: R^3$ → $\displaystyle R^3$ is an orthogonal projection on one of the coordinate

axes, then for every vector x in $\displaystyle R^3$, 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.