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This lecture introduces linear maps (functions between vector spaces that preserve addition and scaling), then defines matrices and shows how matrix-vector multiplication works via the "row times column" rule. The key result: every matrix defines a linear map, establishing a deep connection between algebra and geometry. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] State the definition of a linear map: $T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$
• [ ] Prove a given map is linear (show condition holds for all vectors and scalars)
• [ ] Prove a given map is not linear (find a single counterexample)
• [ ] Understand why linear maps $f: \mathbb{R} \to \mathbb{R}$ must have the form $f(x) = ax$ (no constant term)
• [ ] Know matrix notation: dimensions ($m times n$), element indexing ($a_{ij}$), index form
• [ ] Decompose a matrix into row vectors and column vectors
• [ ] Perform matrix transposition and know $(A^T)^T = A$
• [ ] Compute matrix-vector products using "row times column"
• [ ] Write matrix-vector multiplication in index form: $(Ax)i = \sum_j a{ij} x_j$
• [ ] Know the identity matrix $I_n$ and the Kronecker delta $\delta_{ij}$
• [ ] Express scalar products as matrix operations: $\langle x, y \rangle = x^T y$
• [ ] Know that $A e_i$ projects out the $i$-th column of $A$
• [ ] Prove that every matrix defines a linear map (both explicit 2×2 and index form proofs)

1. Linear Maps — Definition

A map $T: V \to W$ between vector spaces is called linear if:

$$T(alpha x + beta y) = alpha T(x) + beta T(y)$$