## Linear Algebra and Its Applications, Review Exercise 2.30

Review exercise 2.30. Suppose that the matrix $A$ is a square matrix.

a) Show that the nullspace of $A^2$ contains the nullspace of $A$.

b) Show that the column space of $A$ contains the column space of $A^2$.

Answer: a) Suppose $x$ is in the nullspace of $A$, so that $Ax = 0$. We then have $A^2x = A(Ax) = A\cdot 0 = 0$ so that $x$ is also in the nullspace of $A^2$. Since this is true for every $x$ in the nullspace of $A$, the nullspace of $A$ is a subset of the nullspace of $A^2$ or (stated differently) the nullspace of $A^2$ contains the nullspace of $A$.

b) Suppose that $b$ is in the column space of $A^2$. Then there exists some $x$ for which $A^2x = b$ (in other words, $b$ is a linear combination of the columns of $A$, with the entries of $x$ being the coefficients).

Let $y = Ax$. Then we have $Ay = AAx = A^2x = b$. Since there exists a $y$ for which $Ay = b$ we see that $b$ is also in the column space of $A$. Since this is true for every $b$ in the column space of $A^2$, the column space of $A^2$ is a subset of the column space of $A$ or (stated differently) the column space of $A$ contains the column space of $A^2$.

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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