## 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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