How many restrictions?
A model for yield has an intercept, a coefficient for nitrogen, a coefficient for phosphorus, and a coefficient for their product. State R and r for each hypothesis. (a) Nitrogen and phosphorus do not interact. (b) Phosphorus has no effect at any nitrogen level, and there is no interaction. (c) None of the three covariates matters.
Order the coefficients as (\beta_0,\beta_N,\beta_P,\beta_{NP}). (a) No interaction is \beta_{NP}=0, so r=1 and R=(0,0,0,1). (b) No phosphorus effect at any nitrogen level means \beta_P=\beta_{NP}=0, so r=2 and
R=\begin{pmatrix}0&0&1&0\\0&0&0&1\end{pmatrix}.
The nitrogen coefficient is still free. (c) None of the covariates matters is \beta_N=\beta_P=\beta_{NP}=0, so r=3 and R is the 3\times 4 matrix that drops the intercept column.