I'm reviewing for my final exam and one of the practice problems given to me by my professor is this:
Consider the matrix $A = uv^T$, where $u \in \mathbb{R}^{m \times 1}$ and $v \in \mathbb{R}^{n \times 1}$. Find an orthogonal projector onto the range of $A$. Find an orthogonal projector onto the null space of $A$.
We discussed in class that, if $A = QR$ (the QR decomposition), then $QQ^T$ and $I - QQ^T$ are projectors where $QQ^T$ projects onto the range of $A$ and $I - QQ^T$ projects onto $\mathcal{R}(A)^{\perp}$. But I'm still confused about what it means for something to project onto the range of a matrix and don't really know how to come up with the QR decomposition of $A$ since I'm just given that $A = uv^T$.
Any help would be greatly appreciated. Thanks in advance!
In a situation as elementary as this, you can really just go back to the basic definitions: $$ \mathcal{R}(A) := \{y \in \mathbb{R}^{n \times 1} : y = Ax \:\text{for some $x \in \mathbb{R}^n$}\},\\ \mathcal{N}(A) := \{x \in \mathbb{R}^{n \times 1} : Ax = 0\}. $$ Given the construction of $A$, it'll be easy to describe $\mathcal{R}(A)$ as the span of some orthonormal set and $\mathcal{N}(A)$ as the orthogonal complement of the span of some orthonormal set. Once you've done this, just remember that if $S = \operatorname{Span}\{v_1,\dotsc,v_k\}$ for some orthonormal set $\{v_1,\dotsc,v_k\}$ in $\mathbb{R}^{n \times 1}$, then the orthogonal projection onto $S$ is $$ P_S := v_1 v_1^T + \cdots + v_kv_k^T $$ and the orthogonal projection onto $S^\perp$ is $$ P_{S^\perp} = I_n - P_S; $$ the meaning of this is that for any $x \in \mathbb{R}^{n \times 1}$, the orthogonal projection of $x$ onto $S$ is $P_S x$ and the orthogonal projection of $x$ onto $S^\perp$ is $P_{S^\perp}x$.