It turns out that when you select the formula_43 largest singular values, and their corresponding singular vectors from formula_18 and formula_19, you get the rank formula_43 approximation to formula_1 with the smallest error (Frobenius norm). This approximation has a minimal error. But more importantly we can now treat the term and document vectors as a "semantic space". The row "term" vector formula_48 then has formula_43 entries mapping it to a lower-dimensional space dimensions. These new dimensions do not relate to any comprehensible concepts. They are a lower-dimensional approximation of the higher-dimensional space. Likewise, the "document" vector formula_50 is an approximation in this lower-dimensional space. We write this approximation as