The question is: construct a matrix A with $[1, 0, 1]^T$ and $[1, 2, 0]^T$ as a basis for its row space and its column space.
From this I get that it is a 3x3 square matrix with these vectors as rows but I do not know how to cotinue. Could you help me please?
Hint: construct a $3\times3$ matrix such that these vectors are both its rows and columns: $$ \left(\begin{array}{ccc} &1 &1\\ 1& 2 & 0\\ 1 & 0& 1 \end{array} \right) $$ Then fill in the empty element with a number such that the new row/column is linearly dependent on the other rows/colums.
$$\left(\begin{array}{c}x\\1\\1\end{array}\right)=a\left(\begin{array}{c}1\\2\\0\end{array}\right)+b\left(\begin{array}{c}1\\0\\1\end{array}\right)\implies a= \frac12,\quad b=1\implies x=\frac32$$
Consider this matrix $$ \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 2 \\ 0 & 2 & -1 \\ \end{bmatrix} $$ Rank of this matrix is $2$. I just tried to make a symmetric matrix with the two vectors you have as basis for the row space.
