Dimension of the direct product of two vector spaces

Question

The direct product of the vector space $\mathcal V$ of order n and the vector space W of order m is the vector space of order n+m on the set of vectors $\{(v_1,\dots,v_n,w_1,\dots,w_m:\textbf v\in V,\textbf w \in W\}$. Note that $\dim(V \text{direct product} W)=\dim (V)+\dim (W)$.

Why is this? It looks like if we have n=m=3, we have 3x3=9 dimension. (I know this is not possible with a vector of length 6.) Consider a set of three lin indep vectors $\textbf v_1,\textbf v_2,\textbf v_3$ in V and $\textbf w_1,\textbf w_2,\textbf w_3$ in W. Appending $\textbf v_1,\textbf v_2,\textbf v_3$ to $\textbf w_1$, we get three lin indep vectors compared to appending them to $\textbf w_2,\textbf w_3$. Why doesn't this argument work?

Answer 1

The argument doesn’t work because those vectors would be linearly dependent. For example,

$$\begin{pmatrix}\textbf v_1\\\textbf w_1\end{pmatrix}= \begin{pmatrix}\textbf v_1\\ \textbf w_2\end{pmatrix}+ \begin{pmatrix} \textbf v_2\\ \textbf w_1\end{pmatrix}-\begin{pmatrix} \textbf v_2\\ \textbf w_2\end{pmatrix}$$

and hence there is redundancy.