{'fraction_non_alphanumeric': 0.024229074889867842, 'fraction_numerical': 0.011013215859030838, 'mean_word_length': 4.352941176470588, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Linear Algebra and Optimization

1.1 Two Descriptions of Subspaces Two useful descriptions of subspaces are: 1. Equations. We can describe a subspace S as a set of vectors satisfying certain linear relationships between their entries. 2. Generators. We can describe a subspace S as the span of a set of vectors. For instance, the subspace S of 3-dimensional vectors whose third entry is the sum of the first and second entries can be expressed as either: