SECTION 4.6 RANK
Suppose we have a physical system whose input can be described by 38 variables and
whose output is then described by 35 linear equations involving the 38 input variables. Laws
of physics or engineering tell us that there are exactly three input vectors which produce the
zero output vector, and no one of these three input vectors is a linear combination of the
other two. Is it possible to achieve
any
output vector with the proper inputs?
We can consider the rows of a matrix
A
as vectors, just as we’ve considered the columns.
So, the subspace spanned by the rows of
A
is called .
. . . . . . . .
yes, the
row space
of
A
.
If the matrix
A
is
p
×
q
, then where do the row space and column space of
A
live?
If we do row operations to the matrix
A
and get a new matrix
B
, then the rows of
B
are linear combinations of the rows of
A
. Since linear combinations of linear combinations
are linear combinations, the row space of
B
is contained in the row space of
A
. But row
operations are reversible, so the same argument tells us that the row space of
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 Spring '08
 PAVLOVIC
 Linear Algebra, row space, .

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