Chapter 3.5, Problem 15PS

To determine

The four subspaces that stays same when the first two rows of A are exchanged. Also, to write down the vector in the left nullspace of the new matrix after the row exchange if $v=\left(1,2,3,4\right)$ is in the left nullspace of A.

Yes, the four subspaces and row space stay the same when first two rows of matrix A are exchanged.

The vector in left subspace of the new matrix is: $\left(2,1,3,4\right)$

Given:

Left nullspace of A is $v=\left(1,2,3,4\right)$

Calculation:

Even after the rows of a matrix are exchanged, rows stay themselves and the subspaces whose bases are already pre-determined also stay the same. Therefore, the four subspaces and row space stay the same when first two rows of matrix A are exchanged.

Switch rows of the matrix means the order of the equation is changed, however, the result remains the same as that of old matrix. Thus, if $v=\left(1,2,3,4\right)$is in the column space of the matrix, the vector in the column space of the new matrix would be $\left(2,1,3,4\right)$. Therefore, the vector in left subspace of the new matrix is: $\left(2,1,3,4\right)$