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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Linear Algebra and Its Applications, Exercise 3.1.17

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Linear Algebra and Its Applications, Exercise 3.1.17

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