Linear Algebra and Its Applications · Problem 2.4.33

Vector Spaces — Gilbert Strang

Mathematics · four-fundamental-subspaces

Problem

(Left nullspace) Add the extra column

b

and reduce

A

to echelon form:

\begin{bmatrix} A & b \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 & b_1 \\ 4 & 5 & 6 & b_2 \\ 7 & 8 & 9 & b_3 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 3 & b_1 \\ 0 & -3 & -6 & b_2 - 4b_1 \\ 0 & 0 & 0 & b_3 - 2b_2 + b_1 \end{bmatrix}.

A combination of the rows of

A

has produced the zero row. What combination is it? (Look at

b_3 - 2b_2 + b_1

on the right-hand side.) Which vectors are in the nullspace of

A^T

and which are in the nullspace of

A