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Basis and Dimension

Basis & Dimension

To understand the structure of a vector space, we need to know what creates it (Spanning) and what is redundant (Dependence).

Core Concepts

  1. Linear Combination: Creating new vectors by scaling and adding existing ones (c1v1+...+ckvkc_1v_1 + ... + c_kv_k).
  2. Linear Independence: No vector in the set can be written as a linear combination of the others.
  3. Spanning Set: The set of all possible linear combinations.
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The Basis

A Basis is the “Goldilocks” set of a vector space. It is:

  1. Large enough to Span the space.
  2. Small enough to be Linearly Independent .

Dimension

The dimension of a vector space is simply the number of vectors in its basis. This number is unique for a given space.