We consider the task of testing whether a Boolean function $f:\bitset^\ell\to\bitset$ is the indicator function of an $(\ell-k)$-dimensional affine space. An optimal tester for this property was presented by Parnas, Ron, and Samorodnitsky (SIDMA, 2002), by mimicking the celebrated linearity tester (of Blum, Luby and Rubinfeld (JCSS, 1993)) and its analysis. We show that the former task (i.e., testing $(\ell-k)$-dimensional affine spaces) can be reduced to testing the linearity of a related function $g:\bitset^\ell\to\bitset^k$, yielding an almost optimal tester.
Addendum (20/5/16): In addition, we show that testing monomials can be performed by using the foregoing reduction and reducing part of the analysis to the analysis of the dictatorship test. Specifically, in the original post we only treated the question of testing affine spaces, which is the main module in PRS's tester for monomials. In that post, we showed that testing whether $f:\{0,1\}^\ell\to\{0,1\}$ describes an $(\ell-k)$-dim affine space reduces to testing whether a related function $g:\{0,1\}^\ell\to\{0,1\}^k$ is linear. In the this revision, we show that testing $k$-monomials can be performed by combining a tester for affine spaces with a test of $g$ that generalizes the other module in the PRS tester (i.e., the "conjunction test").
Major revision (14/3/20): The current version is the result of an extensive revision. In particular, the reduction of testing affine spaces to testing linearity (of functions) is simplified and extended to arbitrary finite fields, the second step in the procedure for testing $k$-monomials is revised, many of the technical justifications are elaborated, and some crucial typos are fixed. In addition, the title has been augmented for clarity, the brief introduction has been expanded, and the high level structure has been re-organized.