Definition: The size (or dimension) of a vector (\mathbf{v} \in \mathbb{F}^n) is the number (n) of its components.
If (\mathbf{v} = (v_1, v_2, \ldots, v_n)), then the size of (\mathbf{v}) is (n).
Notation: The size of (\mathbf{v}) is often denoted as:
(\text{size}(\mathbf{v}) = n)
(\mathbf{v} \in \mathbb{F}^n) (the superscript indicates the size)
Examples:
(\mathbf{v} = (3, -1, 0, 5)) has size 4, so (\mathbf{v} \in \mathbb{R}^4)
(\mathbf{w} = (1+i, 2-i)) has size 2, so (\mathbf{w} \in \mathbb{C}^2)
Important: Two vectors can only be added if they have the same size. Vector operations are defined component-wise, so mismatched sizes make the operation undefined.
**Definition:** The *size* (or *dimension*) of a vector \(\mathbf{v} \in \mathbb{F}^n\) is the number \(n\) of its components.
If \(\mathbf{v} = (v_1, v_2, \ldots, v_n)\), then the size of \(\mathbf{v}\) is \(n\).
**Notation:** The size of \(\mathbf{v}\) is often denoted as:
- \(\text{size}(\mathbf{v}) = n\)
- \(\mathbf{v} \in \mathbb{F}^n\) (the superscript indicates the size)
**Examples:**
- \(\mathbf{v} = (3, -1, 0, 5)\) has size 4, so \(\mathbf{v} \in \mathbb{R}^4\)
- \(\mathbf{w} = (1+i, 2-i)\) has size 2, so \(\mathbf{w} \in \mathbb{C}^2\)
**Important:** Two vectors can only be added if they have the same size. Vector operations are defined component-wise, so mismatched sizes make the operation undefined.
