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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