Vector operations

Vectors of the same length are added element by element: $\mathbf{u}+\mathbf{v}=(u_1+v_1,\dots ,u_d+v_d)$. Multiplying by a scalar scales every entry: $c\mathbf{v}=(c{v}_1,\dots ,c{v}_d)$. Geometrically, addition places arrows tip to tail and scaling stretches or flips an arrow.

A linear combination is $c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2+\cdots +c_k{\mathbf{v}}_k$. A linear model's prediction is exactly this: a weighted combination of features. The set of all linear combinations of some vectors is their span.

The length (norm) of a vector is $\Vert \mathbf{v}{\Vert }_2=\sqrt{v_1^2+\cdots +v_d^2}$. Dividing by it gives a unit vector pointing in the same direction. Vectors are linearly independent if none of them is a combination of the others.