Direct Product Vector Space

Last updated Nov 1, 2022

Let $V$, $W$ be Vector Spaces over Field $F$. Then $V \times W$ equipped with

1. Additive Identity $(0, 0)$.
2. $+$ defined as $(x_{1}, x_{2}) + (y_{1}, y_{2}) = (x_{1} + x_{2}, y_{1} + y_{2})$ for $(x_{1}, x_{2}), (y_{1}, y_{2}) \in V \times W$.
3. $*$ defined as $c (x_{1}, x_{2}) = (cx_{1}, cx_{2})$ for $c \in F$ and $(x_{1}, x_{2}) \in V \times W$.

forms a Vector Space. It is known as the Direct Product Vector Space

TODO - just run through the Vector Space Axioms.