Linear Algebra 5 | Vector Space ℝn [dark version]

The core takeaway is that 3n (all n-tuples of real numbers) becomes a vector space once vector addition and scalar multiplication are defined component-by-component, and that this structure is guaranteed by eight precise algebraic rules. The construction starts with 3n as the Cartesian product of 3 with itself n times, so each vector is written as a column 3 = (v1, v2, 3, vn) with real entries. Addition of two vectors u and v is defined by adding corresponding components (u1+v1, u2+v2, 3, un+vn). Scalar multiplication uses a real number bb to scale a vector u by multiplying every component by bb: (bb u1, bb u2, 3, bb un). With these operations in place, 3n is not just a set of tuples—it supports the same kind of calculations as familiar spaces like 32 and 33, but in any dimension n.

To make “vector space” more than a label, the transcript lays out the eight defining properties that 3n satisfies. The first four properties concern addition: it is associative, meaning the grouping of sums doesn’t change the result; it has an additive identity, the zero vector (0,0,3,0), which leaves any vector unchanged; every vector has an additive inverse, written as -v = (-v1,-v2,3,-vn), which cancels v back to the zero vector; and addition is commutative, so u+v = v+u. Together, these four properties mean addition forms an abelian group.

The remaining four properties govern scalar multiplication and its interaction with addition. Scalar multiplication must be compatible with multiplication of real numbers: scaling by bc and then by bb matches scaling once by the product bbbc. Scaling by 1 leaves vectors unchanged: 1v = v. Finally, two distributive laws connect scaling to addition. One law distributes a scalar over vector addition: bb(v+w) = bbv + bbw. The other law distributes scalar addition over scaling: (bb+bc)v = bbv + bcv. These rules are what make 3n a vector space in a fully formal sense.

The transcript then highlights canonical unit vectors e1 through en, where ej has a 1 in the j-th position and zeros elsewhere. Their purpose is practical: every vector v in 3n can be written as a linear combination of these unit vectors, specifically v = v1 e1 + v2 e2 + 3 + vn en. This representation ties the abstract vector-space axioms back to concrete computation—components become coefficients, and the unit vectors act as a basis for rebuilding any vector in n-dimensional real space.