## 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 .

This entry was posted in linear algebra and tagged . Bookmark the permalink.