Question

this is linear algebra question In R^3 the orthogonal projections on the x-axis, y-axis, and z-axis are defined by respectively. T1(x,y,z)=(x,0,0), T2(x,y,z)=(0,y,0), 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 T:R^3toR^3 is an orthogonal projection on one of the coordinate axes, then for every vector x in R^3,the vectors 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.

Answer 1

a) show for each of the projections: T(0)=0, T(ax)=aT(x) and T(x+y)=T(x)+T(y), with x and y vectors and a a constant. if you do that you've shown linearity.

for the matrices column one is T((1,0,0)), column 2 is T((0,1,0)) and the last one is T((0,0,1))

Answer 2

thank you sir what about part b and c

Answer 3

I'm not sure how to prove b, I guess showing that the dot product of T(x) and x-T(x) equals zero. But if you have that, then C will be easy: draw a random vector x, draw its projections T(x) on x-axis and finally draw x-T(x) "starting" in T(x) and "ending" in x.

Answer 4

thanks thanks sir i am totally satisfied i have come up with solution by myself