Combining Vectors (Addition & Scaling)

What happens if you and your friend decide to shop together and combine your carts into a single shopping trip? You add your vectors together position-by-position:

x_{\text{total}} = x_{\text{you}} + x_{\text{friend}} = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} + \begin{bmatrix} 0 \\ 5 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 + 0 \\ 3 + 5 \\ 1 + 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 8 \\ 1 \end{bmatrix}

Your combined cart now contains 2 Apples, 8 Bananas, and 1 Carrot.

What if you want to stock up on groceries for 3 weeks instead of 1? You scale your entire list by multiplying the vector by a single number (called a scalar):

3 \cdot x_{\text{you}} = 3 \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \times 2 \\ 3 \times 3 \\ 3 \times 1 \end{bmatrix} = \begin{bmatrix} 6 \\ 9 \\ 3 \end{bmatrix}

Formal Definition ( $n$ -Dimensions)

For any two vectors $u, v \in \mathbb{R}^n$ and any scalar $c \in \mathbb{R}$ , addition and scalar multiplication are defined position-wise:

u + v = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix} + \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{bmatrix} \qquad\qquad c \cdot v = c \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \begin{bmatrix} c v_1 \\ c v_2 \\ \vdots \\ c v_n \end{bmatrix}

Vector Addition: Combines two vectors position by position.
Scalar Multiplication: Stretches or shrinks every entry by the same scalar factor $c$ .
A combination like $2u - v$ is simply these two operations composed: scale $u$ by $2$ , then add $-v$ .

TEASER: How Image Generators Blend Concepts Modern AI image generators use this exact principle in high-dimensional feature spaces: take the feature vector representing a dog, add the feature vector for sunglasses, and the model generates an image of a dog wearing sunglasses.

Combining Vectors (Addition & Scaling)

Formal Definition (nnn-Dimensions)

Formal Definition ( $n$ -Dimensions)