Download test 4

Linear Algebra test four
due Tuesday, March 16, 2004
please show your own independent work to get full credit for each problem
(1) Given the matrix A =
17
6
6
8
(a) Find the eigenvalues & eigenvectors of the matrix.
(b) Diagonalize the matrix A: find an invertible matrix P and a diagonal
matrix D so that A = P DP −1 .
(c) Find an orthogonal matrix P so that D = P T AP .
(d) Rewrite the quadratic form 17x1 2 + 12x1 x2 + 8x2 2 in the format ~xT A~x.
(e) Find a change of variable matrix P so that ~x = P ~y and in the new coordinates y1 and y2 the quadratic form of part (d) has no cross-product
term.
(f) Use parts (d) and (e) to graph 17x1 2 + 12x1 x2 + 8x2 2 = 4, showing all the
relevant axes and units.
(2) Another 2 × 2 matrix B has an eigenvalue λ = 1 with associated eigenvector
~v = (4, 1) and eigenvalue λ = −1 with associated eigenvector w
~ = (−1, 4). Find
a formula for the matrix B, and describe the geometric effect of multiplication
by B.
(3) Calculate the projection of the vector ~b = (5, 1, −3) onto the plane spanned
by the vectors ~v = (2, 2, 1) and w
~ = (−1, 2, 2), and calculate the distance of the
tip of the vector ~b from the plane.
(4) Calculate the projection of the vector ~b = (5, 1, −3, 2) onto the subspace
W = {(x1 , x2 , x3 , x4 ) : x1 + 2x3 − x4 = 0 & x2 − x3 + x4 = 0}
(5) Find the 3 × 3 matrix R which represents a rotation of π/6 around the axis
of rotation ~v = (1, 2, 3).
(6) Given a length one vector ~v in R2 and the associated 2 × 2 matrix A = ~v~v T ,
describe the eigenvalues and eigenvectors of the matrix A.
(7) Given the function f (x) = 2x and the associated data points (−1, f (−1)),
(0, f (0)), and (1, f (1)),
(a) Find the least-squares line y = mx + b which approximates the data, and
calculate the least-squares error.
(b) Find the least squares approximation of the form
y =a+
b
x−2
which approximates the data, and calculate the least-squares error.