Product (Vector Spaces)

definitionvector-spacesset-theory

Definition

Let $V_1,\dots ,V_n$ be vector spaces over $\mathbb{F}$. The product is

$$V_1\times \cdots \times V_n=\left\{(v_1,\dots ,v_n):v_i\in V_i,1\le i\le n\right\}.$$

Addition and scalar multiplication are component-wise:

• $(u_1,\dots ,u_n)+(v_1,\dots ,v_n)=(u_1+v_1,\dots ,u_n+v_n)$.
• $\lambda (u_1,\dots ,u_n)=(\lambda u_1,\dots ,\lambda u_n)$.

Examples