# Thread: Orthogonal Projections and Linear Transformations

1. ## Orthogonal Projections and Linear Transformations

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
(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.

2. 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
(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.

I bet you've already done some self work on both these problems, so show it to us and tell us where're you stuck and somebody will perhaps try to help you.

Tonio

3. Honestly, I don't know 1.) very much.

2.) I know for a.) I used:

Ta(Ta^(-1)(x)) = AA^(-1)x = /x = x on two one-to-one linear transformations. I used a:

X 0
0 X

matrix to see if I get the same equation back. Same principle for b.)