Homework 5
MATH 132
Prof. Janet Vassilev
January 29, 2007
1. Determine if T:R3โR3given by T(x, y, z) = (z , x โy, x +y) is orthogonal.
2. Find the projection matrix of R3onto the subspace sp{(1,2,1),(2,0,โ3)}.
3. Let Abe an mรnmatrix. Let Qbe the projection of Rnonto the row space of A.
Show that IโQis the projection of Rnonto N(A).
January 31, 2007
4. Find the projection matrix of R4onto the subspace spanned by ( 5
13 ,0,12
13 ,0) and
(5
13 ,0,โ12
13 ,0).
5. Find the projection matrix of R3onto the subspace spanned by ( โ3
3,โ3
3,โโ3
3) and
(โ2
2,0,โ2
2).
6. Let Pbe an nรnmatrix such that P2=Pand every vector in N(P) is orthogonal
to every vector in the column space of P. Show Pis a projection matrix.
February 2, 2007
7. In a lab, some data is collected.
(1,10),(2,7),(3,5),(4,1)
Theoretically, the data should fit a linear equation. Find the linear equation.
8. Find the least squares solution for the system of equations.
2x+y= 7
xโy= 4
3x+ 2y= 9
1