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:
- \((\mathbf{u} + \mathbf{v})_j = u_j + v_j\)
- \((a\mathbf{v})_j = a \cdot v_j\)Copy