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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