## 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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