Linear Algebra and Its Applications, Exercise 3.3.20

Exercise 3.3.20. Given the matrix P_R that projects onto the row space of A, find the matrix P_N that projects onto the nullspace of A.

Answer: The null space of A is orthogonal to the row space of A. The two spaces are orthogonal complements, with \mathcal{N}(A) = \mathcal{R}(A^T)^\perp. Recall from exercise 3.3.11 that if P is a projection matrix onto S and Q a projection matrix onto S^\perp then we have P+Q=I.

So in this case we have P_R + P_N = I or P_N = I - P_R.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.

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