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This lecture builds towards the definition of dimension by introducing the two key ingredients: linear independence (not too many vectors) and span (enough vectors). These combine to define a basis, with the canonical basis as the familiar special case. The lecture also introduces the nullspace/kernel of a matrix and orthonormal bases. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] State the definition of linear dependence: one vector can be written as a linear combination of the others
• [ ] Test linear independence by solving a system of equations (or checking $Ax = 0$)
• [ ] Know the equivalent formulation: vectors are dependent iff $\alpha_1 v_1 + \ldots + \alpha_n v_n = 0$ has a non-trivial solution
• [ ] Define the homogeneous equation $Ax = 0$ and know the zero vector is always a (trivial) solution
• [ ] Define and compute the nullspace (kernel) of a matrix: $\text{Null}(A) = \{x : Ax = 0\}$
• [ ] Connect nullspace to independence: trivial nullspace ↔ columns are independent
• [ ] Know that $n$ vectors in $\mathbb{R}^p$ are always dependent if $n > p$
• [ ] Define the span of a set of vectors as the set of all their linear combinations
• [ ] Determine the geometric shape of a span (line, plane, etc.)
• [ ] Define basis: a linearly independent set that spans the entire space
• [ ] Verify whether a given set forms a basis (check both requirements)
• [ ] State the canonical (standard) basis for $\mathbb{R}^d$ and find coordinates in it
• [ ] Define orthonormal basis: $\langle v_i, v_j \rangle = \delta_{ij}$ (unit length + right angles)
• [ ] Define the dimension of a vector space as the number of vectors in any basis

1. Linear Independence

Motivation: What is Dimension?