January 8

2.1 Vectors

A vector is a list of number with addition and scalar multiplication defined. Given vectors u = (u₁, u₂, …, u_n) ∈ ℝⁿ, v = (v₁, v₂, …, v_n) ∈ ℝⁿ of equal dimension and a scalar c ∈ ℝ, we define * addition: u + v = (u₁ + v₁, u₂ + v₂, …, u_n + v_n), * scalar multiplication: cu = (cu₁, cu₂, …, cu_n).

go over the geometry in class. tail to tip, parallelogram

Let a, b be scalars and u, v, w ∈ ℝⁿ. Then

• u + v = v + u,
• a(u + v) = au + av,
• (a + b)u = au + bu,
• (u + v) + w = u + (v + w),
• a(bu) = (ab)u,
• u + ( − u) = 0,
• u + 0 = 0 + u = u,
• 1u = u.

Definition: The If u₁, u₂, …, u_m are vectors and c₁, c₂, …, c_m are scalars, then
c₁u₁ + c₂u₂ + … + c_mu_m
is a linear combination of u₁, …, u_m. Note that the constants can be negative or zero.