26 Math Castroconde
Chapter 6
1.
2.
3.
Name:
I.D.#:
4
3
1
3
4
, y = , z = 2 are mutually orthogonal and find a nonzero
Show that x =
2
1
3
1
2
4
vector w in R 4 such that {x, y, z w} is an orthogonal set.
1
1
3
Let u = 1 , w = 1 , v = 1 . Let W = span{u, w}.
1
0
1
a.
Show that {u, w} is an orthogonal basis for W.
b.
Find projW v .
c.
Find the distance from v to W.
Let W be the subspace of R 3 represented by the plane with equation x + y + z =0.
1 1
Show that B = 0 , 1 is a basis for W and use it to construct an orthonormal
1 0
basis W.
4.
Find for the orthogonal complement W of the subspace W of question 3.
5.
Assume A and B are n x n orthogonal matrices, and let k be any nonzero real
number. Determine if the matrices kA, A + B, and AB are orthogonal by proving
that each is or by providing a counter example if it is not.
6.
3
1
1
1 , u = 2 , w = 4 , and let W = {v, u, w}.
Let v =
1
1
7
a.
b.
c.
7.
Show that W is an orthogonal set.
Using the elements of W, construct an orthogonal matrix A.
Find A1 .
1 2
3
1 4 , b = 1 . Find a least-squares solution of Ax = b.
Let A =
1 2
5