Definition of entry/component of a vector

Definition: An entry (or component) of a vector (\mathbf{v} = (v_1, v_2, \ldots, v_n)) is one of the individual scalars (v_j) that make up the vector, where (j = 1, 2, \ldots, n).

Notation: The (j)-th component of (\mathbf{v}) is denoted:

• (v_j) — subscript notation
• ((\mathbf{v})_j) — explicit indexing

Examples:

• If (\mathbf{v} = (3, -1, 0)), then (v_1 = 3), (v_2 = -1), (v_3 = 0)
• If (\mathbf{w} = (1+i, 2-i, 4, i)), then (w_1 = 1+i), (w_4 = i)

Component-wise operations: Vector addition and scalar multiplication are defined by operating on each component individually:

• ((\mathbf{u} + \mathbf{v})_j = u_j + v_j)
• ((a\mathbf{v})_j = a \cdot v_j)