Construct the standard matrix of the orthogonal projection o

Construct the standard matrix of the orthogonal projection onto the line Is the orthogonal projection onto the line in R^3 a linear map? Justify briefly.

Solution

b)

The line has direction vector v=(1,2). The projection of (x,y)R onto the line is given by

projv(x,y)=((x,y)v/vv)v=x+2y/5v.

c)   [projv(1,0)  projv(0,1)]=[1/5 2/5 2/5 4/5]=1 /5[1 2 2 4].

The line has direction vector v=(1,2). The projection of (x,y)R onto the line is given by projv(x,y)=((x,y)v/vv)v=x+2y/5v. c)   [projv(1,0)  projv(0,1)]=[1/5 2/5 2/5 4/5]=1 /5[1 2 2 4].