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This lecture introduces abstract vector spaces as generalisations of coordinate spaces, then defines norms (lengths) and scalar products (dot products) to measure distances and angles between vectors. These tools are essential for machine learning, computer graphics, and scientific computing. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Understand the abstract definition of a vector space and its 8 axioms
• [ ] Verify that $\mathbb{R}^n$ satisfies all vector space axioms
• [ ] Know how polynomials can form a vector space (non-examined)
• [ ] Calculate the norm (length) of a vector using the generalised Pythagoras theorem
• [ ] Normalise a vector to unit length
• [ ] Compute the scalar product (dot product) of two vectors
• [ ] Express norms in terms of scalar products: $\|x\| = \sqrt{\langle x, x \rangle}$
• [ ] Use the angle formula: $\cos(\theta) = \langle x, y \rangle / (\|x\| \|y\|)$
• [ ] Identify orthogonal vectors ($langle x, y rangle = 0$)
• [ ] State and apply the Cauchy-Schwarz and Triangle inequalities

1. Abstract Vector Spaces (Non-Examined)

The abstract definition is like object-oriented programming — you define a class with axioms (properties), then develop theory that works for all objects in that class, not just coordinate spaces.

Why Go Abstract?

• Fourier transforms, Hilbert spaces → computer vision, quantum computing
• Develop theory once, apply it in many contexts
• Don't need to remember context-specific details