Linear Algebra, 2020a

Section III. Basis and Dimension 139
Thus, the first point for this subsection is that the tools of this chapter give
us a more conceptual understanding of Gaussian reduction.
For the second point observe that row operations on a matrix can change its
column space. ( )
(
1 2 −2ρ 1 +ρ 2 1 2
−→
2 4
0 0
)
The column space of the left-hand matrix contains vectors with a second component
that is nonzero but the column space of the right-hand matrix contains
only vectors whose second component is zero, so the two spaces are different.
This observation makes next result surprising.
3.10 Lemma Row operations do not change the column rank.
Proof Restated, if A reduces to B then the column rank of B equals the column
rank of A.
This proof will be finished if we show that row operations do not affect linear
relationships among columns, because the column rank is the size of the largest
set of unrelated columns. That is, we will show that a relationship exists among
columns (such as that the fifth column is twice the second plus the fourth) if and
only if that relationship exists after the row operation. But this is exactly the
first theorem of this book, Theorem One.I.1.5: in a relationship among columns,
⎛ ⎞
⎛ ⎞ ⎛ ⎞
a 1,1
a 1,n 0
a 2,1
c 1 ·
⎜
⎝
⎟
. ⎠ + ···+ c a 2,n
n ·
⎜ . ⎝
⎟
. ⎠ = 0
⎜
⎝
⎟
. ⎠
a m,1 a m,n 0
row operations leave unchanged the set of solutions (c 1 ,...,c n ).
QED
Another way to make the point that Gauss’s Method has something to say
about the column space as well as about the row space is with Gauss-Jordan
reduction. It ends with the reduced echelon form of a matrix, as here.
⎛
⎜
1 3 1 6
⎞
⎛
⎟
⎜
1 3 0 2
⎞
⎟
⎝2 6 3 16⎠ −→ ··· −→ ⎝0 0 1 4⎠
1 3 1 6
0 0 0 0
Consider the row space and the column space of this result.
The first point made earlier in this subsection says that to get a basis for the
row space we can just collect the rows with leading entries. However, because
this is in reduced echelon form, a basis for the column space is just as easy: collect
the columns containing the leading entries, 〈⃗e 1 ,⃗e 2 〉. Thus, for a reduced echelon