Final answer:
To construct a matrix whose column space contains (1,1,0) and (0,0,1) and the row space contains (1,2) and (2,5), create a matrix with those columns and scale the second row to match the desired row space.
Step-by-step explanation:
To construct a matrix whose column space contains (1,1,0) and (0,0,1) and the row space contains (1,2) and (2,5), we can start by creating a matrix with these columns:
| 1 0 |
| 1 0 |
| 0 1 |
To ensure that the row space includes (1,2) and (2,5), we need to multiply the second row of the matrix by a scalar to match the second row of the desired row space. We can achieve this by scaling the second column:
| 1 0 |
| 2 0 |
| 0 1 |
This matrix satisfies the conditions given, with (1,1,0) and (0,0,1) as the column space and (1,2) and (2,5) as the row space.