Linear Algebra and Its Applications, Exercise 3.1.17

Exercise 3.1.17. Suppose that V and W are subspaces of \mathbb{R}^n and are orthogonal complements. Is there a matrix A such that the row space of A is V and the nullspace of A is W? If so, show how to construct A using the basis vectors for V.

Answer: Suppose v_1 through v_m are the basis vectors for V. (Since V is a subspace of \mathbb{R}^n we will have m \le n.) Let A be an m by n matrix with row 1 equal to v_1, row 2 equal to v_2, and so on through row m equal to v_m.

The row space of A is the set of all linear combinations of the rows, which is equivalent to the set of all linear combinations of v_1 through v_m. But since v_1 through v_m is a basis set for V they span V and the set of all linear combinations of v_1 through v_m is V itself. The row space of A is therefore equal to V.

Now consider the null space of A consisting of all vectors x such that Ax = 0. If this is the case then the inner product of x with each row of A must be zero or (put another way) x must be orthogonal to each row of A. Since the rows of A are v_1 through v_m this means that any vector x in the nullspace of A is orthogonal to all of v_1 through v_m.

Since x is orthogonal to all of v_1 through v_m it is also orthogonal to all linear combinations of v_1 through v_m, and since v_1 through v_m form a basis set for V this means that x is orthogonal to any vector in V. All vectors in the nullspace are thus orthogonal to V. Moreover, any vector orthogonal to V (i.e., orthogonal to all vectors in V) must be orthogonal to all of v_1 through v_m and is therefore in the nullspace of A.

The nullspace of A thus contains all vectors orthogonal to V. But the set of all vectors orthogonal to V is W, the orthogonal complement to V. The nullspace of A is therefore equal to W.

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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