Vector Addition

To add two vectors in $\mathbb{R}^m$, simply add their corresponding entries. That's it: $(u_1, \ldots, u_m) + (v_1, \ldots, v_m) = (u_1 + v_1, \ldots, u_m + v_m)$.

Geometrically, vector addition follows the parallelogram rule: place both vectors at the origin, complete the parallelogram, and the sum is the diagonal. Equivalently, place them tip-to-tail.

We can only add vectors of the same size — you can't add a vector in $\mathbb{R}^2$ to one in $\mathbb{R}^3$. This constraint will become important when we think about matrix multiplication.