Subspaces and Maps to Other Spaces

Status: outline — bridge span/dependence notebooks to linear maps.

Goals¶

• Vector spaces and subspaces (span, basis, dimension — recap)

• A matrix as a map (T: \mathbb{R}^n \to \mathbb{R}^m)

• Image (range) and kernel (null space)

• Change of coordinates and embedding into other feature spaces

Outline¶

1. Subspaces defined by span and linear equations

2. Column space vs null space

3. Linear maps: projection, rotation, scaling

4. Data view: each feature map sends points to a new space

5. Lab pointers: Linear-Dependence-and-Span, Projection, Multiplying-Matrices-and-Vectors

Sources¶

• linear-algebra-basics.md and the linear-algebra notebooks

• Goodfellow et al., Deep Learning, Ch. 2 (as used in these notes)