Math 1080:
Homework #2
Problem 1:
Find the orthogonal projector P onto range(A) where
𝐴 = [
1 −1
0 2
1 1
]
What is the nullspace of P? What is the image under P of the vector [3 3 0]
𝑇
?
Problem 2:
Let A be 𝑚 × 𝑛 matrix with 𝑚 > 𝑛 , and let
A = Q
ˆ
R
ˆ
be a reduced QR factorization. Show that
A has full rank if and only if all the diagonal entries of
R
ˆ
are nonzero.
Problem 3:
Using Gram-Schmidt orthogonalization compute the QR factorization of the following matrix
𝐴 = [
1 0 2
−2 3 −4
−2 6 5
]
Problem 4: Show that if P is a projector, then ‖𝑃‖2 ≥ 1.
(Hint: For (a), take an arbitrary vector and decompose as 𝑥 = 𝑃𝑥 + (𝐼 − 𝑃)𝑥. Use the triangle
inequality to conclude that ‖𝑥‖2 ≤ ‖𝑃𝑥‖2 . Now use the definition of ‖𝑃‖2 to conclude the
result.)
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