Linear Algebra and Its Applications, Exercise 3.1.13

Exercise 3.1.13. Provide a picture showing the action of A^T in sending the column space of A to the row space and the left nullspace to zero.

Answer: I’m leaving this post as a placeholder until I have time to illustrate this.

The basic idea is that just as the row space and the nullspace are orthogonal complements, so are the column space and the left nullspace. Thus a given vector y should be decomposable into a column space component y_c and a left nullspace component y_{ln}.

Multiplying y_{ln} by A^T produces A^Ty_{ln} = 0 since y_{ln} is in the left nullspace and satisfies yA = 0 = A^Ty. Multiplying y_c by A^T produces A^Ty_c, which is a linear combination of the columns of A^T. But the columns of A^T are the rows of A, so A^Ty_c is a linear combination of the rows of A and is therefore in the row space of A.

So multiplication by A^T sends all vectors in the left nullspace of A into zero, and all vectors in the column space of A into the row space of 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, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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