Dot product

Notation:

$$x^{T}y=x_1{y}_1+\cdots +x_n{y}_n$$

Computational cost: $O(n)$.

Two vectors $x$ and $y$ are orthogonal if $x^{T}y=0$.

For a subspace $S$, we can define the orthogonal complement $S^{\perp }$ as the subspace of vectors $y$ such that:

$$S^{\perp }=\{y|\forall x\in S,x^{T}y=0\}$$

We have: $\text{dim}(S)+\text{dim}(S^{\perp })=n$.

Vectors and matrices