We are given a linear operator
Where is n-dimensional real vector space.
Thus, there is a standard matrix corresponding to the transformation.
Let be the ordered basis.
We are told that the first vector is an eigenvector of the transformation. That is,
Now relative to the ordered basis, its coordinate vector is,
Finally, the standard matrix for the linear operator is given by,
But the first coloum in this matrix is,
The coordinate vector relative to the ordered basis.
Which we found to be,
Thus the first entry is and everything else is zero.