Math for Machine Learning - Chapter 2: Advanced Vector Spaces & Subspaces

This quiz is designed for an expert-level assessment of Vector Spaces and Subspaces, fundamental concepts in linear algebra that form the bedrock of many machine learning algorithms. It goes beyond basic definitions to test deep conceptual understanding, the ability to verify axioms in non-standard spaces, and the skill to identify subtle properties of span, linear independence, and subspaces. These concepts are the 'playground' for data vectors, model parameters, and error vectors in ML. A strong grasp is crucial for understanding topics like dimensionality reduction (PCA), regularization, and the geometry of model solution spaces.

Key Concepts Tested:

• Vector Space Axioms: A set V is a vector space over a field F (e.g., $ℝ$) if for any $u, v, w \in V$ and scalars $c, d \in F$, it satisfies ten axioms including closure, associativity, commutativity, identity and inverse elements, and distributivity.

• Subspace Criteria: A non-empty subset $W$ of a vector space $V$ is a subspace if and only if it is closed under vector addition ($u, v \in W \implies u+v \in W$) and scalar multiplication ($v \in W \implies c v \in W$).

• Linear Combination & Span: A vector $v$ is a linear combination of $v_1, ..., v_k$ if $v = \sum_{i=1}^{k} c_i v_i$. The span of a set of vectors is the set of all their possible linear combinations, which always forms a subspace.

• Linear Independence: A set of vectors ${v_1, ..., v_k}$ is linearly independent if the only solution to $\sum_{i=1}^{k} c_i v_i = 0$ is $c_1 = c_2 = ... = c_k = 0$.