the column space is very easy to understand, for example, we have:
$$\begin{pmatrix}
1& 0& 0 \\
0& 1& 0\\
0& 0& 1
\end{pmatrix}\times \begin{pmatrix}
a \\ b\\ c
\end{pmatrix}$$
if we let $V_1=\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}, V_2=\begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix},V_3=\begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix}$, so the product of above will be$$ \begin{pmatrix} V_1& V_2& V_3 \end{pmatrix}\times\begin{pmatrix}a \\ b\\ c\end{pmatrix}$$, so any vector in $R^3$ can be seen as $aV_1+bV_2+cV_3$
but for row space, I don't know how can I do the same thing, for example,if we set $V_1=\begin{pmatrix} 1& 0& 0 \end{pmatrix}, V_2=\begin{pmatrix} 0& 1& 0 \end{pmatrix},V_3=\begin{pmatrix} 0& 0& 1 \end{pmatrix}$, then it becomes $$ \begin{pmatrix} V_1\\ V_2\\ V_3 \end{pmatrix}\times\begin{pmatrix}a \\ b\\ c\end{pmatrix}$$, but we cannot even multiply these two matrices,can we? so how can I use row vectors to span $R^3$ in the form as $aV_1+bV_2+cV_3$ where $V_1,V_2,V_3$ are row vectors?
Another question is, are $\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}$ the same as $\begin{pmatrix} 1& 0& 0 \end{pmatrix}$ as they all in $R_3$
Best Answer
I cannot comment since I don't have enough reps for that.
What about $(a\quad b\quad c)\times \begin{pmatrix} V_1\\ V_2\\ V_3 \end{pmatrix}$?
This is 1 x 3 matrix times 3 x 3 which gives you a 1 x 3 matrix.
Post multiplication of a matrix by a vector gives a linear combination of the columns of the matrix.
Pre multiplication of a matrix by a vector gives a linear combination of the rows of the matrix.
In answer to your second question:
$\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}$ is not the same as $\begin{pmatrix} 1& 0& 0 \end{pmatrix}$. A vector or point, by default (unless stated otherwise) is assumed to be a column vector. So, $\begin{pmatrix} 1& 0& 0 \end{pmatrix} = \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}^t$.
The operations you can perform with a row vector are different from those that can be performed by a column vector. Examples of such different operations are exactly the two examples (pre and post multiplication) you have provided in your OP.