## 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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