Linear Algebra

Section III. Basis and Dimension 127
So the instructions for the prior example are “transpose, reduce, and transpose
back”.
We can even, at the price of tolerating the as-yet-vague idea of vector spaces
being “the same”, use Gauss’ method to find bases for spans in other types of
vector spaces.
3.9 Example To get a basis for the span of {x 2 + x 4 , 2x 2 +3x 4 , −x 2 − 3x 4 }
in the space P4, think of these three polynomials as “the same” as the row
vectors � 0 0 1 0 1 � , � 0 0 2 0 3 � , and � 0 0 −1 0 −3 � , apply
Gauss’ method
⎛
0
⎝0 0
0
0
0
1
2
−1
0
0
0
⎞
1
3 ⎠
−3
−2ρ1+ρ2 2ρ2+ρ3
−→ −→
ρ1+ρ3
⎛
0
⎝0 0
0
1
0
0
0
⎞
1
1⎠
0 0 0 0 0
and translate back to get the basis 〈x 2 + x 4 ,x 4 〉. (As mentioned earlier, we will
make the phrase “the same” precise at the start of the next chapter.)
Thus, our first point in this subsection is that the tools of this chapter give
us a more conceptual understanding of Gaussian reduction.
For the second point of this subsection, consider the effect on the column
space of this row reduction.
� �
1 2 −2ρ1+ρ2
−→
2 4
� �
1 2
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 is different
because it contains only vectors whose second component is zero. It is this
knowledge that row operations can change the column space that 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.
We will be done if we can show that row operations do not affect linear relationships
among columns (e.g., if the fifth column is twice the second plus the
fourth before a row operation then that relationship still holds afterwards), because
the column rank is just the size of the largest set of unrelated columns. But
this is exactly the first theorem of this book: in a relationship among columns,
⎛ ⎞
a1,1
⎜ a2,1 ⎟
⎜ ⎟
c1 · ⎜ .
⎝ .
⎟
. ⎠ + ···+ cn
⎛ ⎞
a1,n
⎜ a2,n ⎟
⎜ ⎟
· ⎜ .
⎝ .
⎟
. ⎠ =
⎛ ⎞
0
⎜
⎜0
⎟
⎜.
⎝.
⎟
. ⎠
0
am,1
am,n
row operations leave unchanged the set of solutions (c1,... ,cn). QED