Linear Algebra and Its Applications, Exercise 3.1.15

Exercise 3.1.15. Is there a matrix such that the vector (1, 2, 1) is in the row space of the matrix and the vector (1, -2, 1) is in the nullspace of the matrix?

Answer: The row space of any matrix A is the orthogonal complement to the nullspace of A, so that any vector in the nullspace must be orthogonal to any vector in the row space and vice versa. In other words, the inner products of any such pairs of vectors must be zero.

Suppose that the vector (1, 2, 1) is in the row space of some matrix A. The inner product of (1, 2, 1) with (1, -2, 1) is

1 \cdot 1 + 2 \cdot (-2) + 1 \cdot 1 = 1 - 4 + 1 = -2

Since the inner product of the two vectors is not zero, the vector (1, -2, 1) cannot be in the nullspace of A.

Similarly, assume that (1, -2, 1) is in the nullspace of some matrix B. Then because the inner product with (1, 2, 1) is nonzero, the vector (1, 2, 1) cannot be in the row space of B.

We conclude that no matrix exists for which (1, 2, 1) is in the row space and (1, -2, 1) is in the nullspace.

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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