## Linear Algebra and Its Applications, Exercise 2.4.5

Exercise 2.4.5. Suppose that $AB = 0$ for two matrices $A$ and $B$. Show that the column space $\mathcal R(B)$ is contained within the nullspace $\mathcal N(A)$ and that the row space $\mathcal R(A^T)$ is contained within the left nullspace $\mathcal N(B^T)$.

Answer: Assume that $A$ is an $m$ by $n$ matrix and $B$ is an $n$ by $p$ matrix, and that $v_1, v_2, \dotsc, v_p$ are the columns of $B$. Since each row of $A$ multiplies each column of $B$ to produce zero, we must have $Av_j = 0$ for all $1 \le j \le p$. In other words, each $v_j$ is in the nullspace of $A$.

Let $v$ by any vector in the column space of $B$. We then have

$v = c_1v_1 + c_2v_2 + \cdots + c_pv_p$

for some set of coefficients $c_1, c_2, \dotsc, c_p$ and thus

$Av = Ac_1v_1 + Ac_2v_2 + \cdots + Ac_pv_p$

$= c_1Av_1 + c_2Av_2 + \cdots + c_pAv_p$

$= c_1 \cdot 0 + c_2 \cdot 0 + \cdots c_p \cdot 0 = 0$

So any vector $v$ in the column space of $B$ is also in the nullspace of $A$, and thus $\mathcal R(B)$ is contained within $\mathcal N(A)$.

Similarly, let $w_1, w_2, \dotsc, w_m$ be the rows of $A$. Since each row of $A$ multiplies each column of $B$ to produce zero, we must have $w_iB = 0$ for all $1 \le i \le m$. In other words, each $w_i$ is in the left nullspace of $B$.

Let $w$ by any vector in the row space of $A$. We then have

$w = c_1w_1 + c_2w_2 + \cdots + c_pw_m$

for some set of coefficients $c_1, c_2, \dotsc, c_p$ and thus

$wB = c_1w_1B + c_2w_2B + \cdots + c_pw_pB$

$= c_1 \cdot 0 + c_2 \cdot 0 + \cdots c_p \cdot 0 = 0$

So any vector $w$ in the row space of $A$ is also in the left nullspace of $B$, and thus $\mathcal R(A^T)$ is contained within $\mathcal N(B^T)$.

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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