Page 38 - 35Linear Algebra
P. 38
38 Systems of Linear Equations
Augmented Matrix Notation
Another interesting rewriting is
1 1 27
x + y = .
2 −1 0
1
This tells us that we are trying to find the combination of the vectors and
2
1 27 1 1
adds up to ; the answer is “clearly” 9 + 18 .
−1 0 2 −1
Here is a larger example. The system
1x + 3y + 2z + 0w = 9
6x + 2y + 0z − 2w = 0
−1x + 0y + 1z + 1w = 3 ,
is denoted by the augmented matrix
1 3 2 0 9
6 2 0 −2 0 ,
−1 0 1 1 3
which is equivalent to the matrix equation
x
1 3 2 0 9
y
6 2 0 −2 = 0 .
z
−1 0 1 1 3
w
Again, we are trying to find which combination of the columns of the matrix
adds up to the vector on the right hand side.
For the the general case of r linear equations in k unknowns, the number
of equations is the number of rows r in the augmented matrix, and the
number of columns k in the matrix left of the vertical line is the number of
unknowns, giving an augmented matrix of the form
1 1 1 1
a a · · · a b
1 2 k
2 2 2 2
a 1 a 2 · · · a k b
. . . . . . . . .
. . . .
a r 1 a r 2 · · · a r k b r
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