Linear Algebra and Its Applications · Problem 2.4.34

Vector Spaces — Gilbert Strang

Mathematics · four-fundamental-subspaces

Problem

Following the method of Problem 33, reduce

A

to echelon form and look at zero rows. The

b

column tells which combinations you have taken of the rows: (a)

\begin{bmatrix} 1 & 2 & b_1 \\ 3 & 4 & b_2 \\ 4 & 6 & b_3 \end{bmatrix}

. (b)

\begin{bmatrix} 1 & 2 & b_1 \\ 2 & 3 & b_2 \\ 2 & 4 & b_3 \\ 2 & 5 & b_4 \end{bmatrix}

. From the

b

column after elimination, read off

m - r

basis vectors in the left nullspace of

A

(combinations of rows that give zero).