Final answer:
The statement is false because a vector n being orthogonal to all data points does not ensure zero cost; it simply means that the vector is perpendicular to the data space.
Step-by-step explanation:
The assertion that β achieves zero cost if and only if there exists some n in the subspace orthogonal to all the data is false. In the context of machine learning or vector calculus, zero cost typically refers to the situation where a prediction made by a model perfectly matches the observed data. However, having a vector n that is orthogonal to all data points does not guarantee a zero cost unless n is specifically the difference between the predicted values and actual values and that difference is zero, which implies perfect prediction. For a vector orthogonal to all data points, it only means that n is perpendicular to the vector space spanned by the data, not that it leads to a zero cost.