## Linear Algebra and Its Applications, Exercise 3.3.8

Exercise 3.3.8. Suppose that $P$ is a projection matrix from $\mathbb R^n$ onto a subspace $S$ with dimension $k$. What is the column space of $P$? What is its rank?

Answer: Suppose that $b$ is a arbitrary vector in $\mathbb R^n$. From the definition of $P$ we know that $Pb$ is a vector in $S$. But $Pb$ is a linear combination of the columns of $P$, so that $Pb$ also is in the column space $\mathcal{R}(P)$. Since any vector in $S$ can be expressed as $Pb$ for some $b$,  all vectors in $S$ are also in $\mathcal{R}(P)$, so that $S \subseteq \mathcal{R}(P)$.

Now suppose that $v$ is an arbitrary vector in the column space $\mathcal{R}(P)$. Then $v$ can be expressed as a linear combination of the columns of $P$ for some set of coefficients $a_1, a_2, \dots, a_n$. Consider the vector $w = (a_1, a_2, \dots, a_n)$. We then have $v = Pw$ by the definition of $w$. But if $v = Pw$ for some $w$ then $v$ is in $S$. So all vectors in $\mathcal{R}(P)$ are also in $S$ and thus $\mathcal{R}(P) \subseteq S$.

Since $S \subseteq \mathcal{R}(P)$ and $\mathcal{R}(P) \subseteq S$ we then have $S = \mathcal{R}(P)$: The column space of $P$ is $S$.

The rank of $P$ is the dimension of its column space. But since $S$ is the column space of $P$, the rank of $P$ is $k$, the dimension of $S$.

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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