Bilinear

Definition. Bilinear If $V,W,\text{and}Z$ are vector spaces, a map $B:V\times W\to Z$ is said to be bilinear if it is linear in each variable sparately when

B(a_{1}v_{1} + a_{2}v_{2}, w) = a_{1}B(v_{1}, w) + a_{2}B(v_{2}, w)\\ B(v, a_{1}w_{1} + a_{2}w_{2}) = a_{1}B(v, w_{1}) + a_{2}B(v, w_{2}) \end{align}$$ --- # Algebra over $\mathbb{R}$ **Definition.** *Algebra* An **algebra** is a real vector space $V$ endowed with a bilinear product map $V \times V \to V$ ## Remark The algebra is said to be ***commutative*** or ***associative*** if the bilinear product has that property.