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 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
.
What is the row space of
A
a subspace of?
If we do row operations to
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, Algebra, Equations, row space

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