University:
Massachusetts Institute of Technology
Course:
18.06SC | Linear Algebra
Academic year:

2021
Views:

320

Pages:

1
Author:

Ireland Shannon

Exercises on projections onto subspaces Problem 15.1: (4.2 #13. Introduction to Linear Algebra: Strang) Suppose A is the four by four identity matrix with its last column removed; A is four by three. Project b = (1, 2, 3, 4) onto the column space of A. What shape is the projection matrix P and what is P? Solution: P will be four by four since we are projecting a 4-dimensional vector to another 4-dimensional vector. We will have: ⎡ ⎤ 1 0 0 0 ⎢ 0 1 0 0 ⎥ ⎥ P=⎢ ⎣ 0 0 1 0 ⎦ 0 0 0 0 This can be seen by observing that the column space of A is the wxy−space, so we just need to subtract the z coordinate from the 4-dimensional vector (w, x, y, z) we’re projecting. The projection of b is therefore: ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 0 0 0 1 1 ⎢ 0 1 0 0 ⎥⎢ 2 ⎥ ⎢ 2 ⎥ ⎥⎢ ⎥ ⎢ ⎥ p = Pb = ⎢ ⎣ 0 0 1 0 ⎦⎣ 3 ⎦ = ⎣ 3 ⎦. 0 0 0 0 4 0 Problem 15.2: (4.2 #17.) If P2 = P, show that ( I − P)2 = I − P. For the matrices A and P from the previous question, P projects onto the column space of A and I − P projects onto the . Solution: ( I − P)2 = I 2 − IP − PI + P2 = I − 2P + P2 = I − 2P + P = I − P. Using the matrices A and P from the previous question, ⎡ ⎤ 0 0 0 0 ⎢ 0 0 0 0 ⎥ ⎥ I−P=⎢ ⎣ 0 0 0 0 ⎦ 0 0 0 1 projects onto the left nullspace of A. 1