## Linear Algebra and Its Applications, Exercise 3.3.19

Exercise 3.3.19. Given a matrix $A$, the matrix $P_C = A(A^TA)^{-1}A^T$ projects onto the column space of $A$. Find the matrix $P_R$ that projects onto the row space of $A$.

Answer: The row space of $A$ is the column space of $A^T$. We can then find the matrix that projects onto $\mathcal{R}(A^T)$ using the standard formula but substituting $A^T$ for $A$: $P_R = A^T[(A^T)^TA^T]^{-1}(A^T)^T = A^T(AA^T)^{-1}A$

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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