Linear Algebra, 2020a

Topic
Cramer’s Rule
A linear system is equivalent to a linear relationship among vectors.
( ) ( ) ( )
x 1 + 2x 2 = 6
1 2 6
⇐⇒ x 1 · + x 2 · =
3x 1 + x 2 = 8
3 1 8
In the picture below the small parallelogram is formed from the vectors ( )
1
3
and ( (
2
1)
. It is nested inside a parallelogram with sides
1
(
x1
3)
and
2
x2
1)
. By the
vector equation, the far corner of the larger parallelogram is ( 6
8)
.
( 6
8)
( 1
x 1 ·
3)
( 1
3)
( 2
1)
( 2
x 2 ·
1)
This drawing restates the algebraic question of finding the solution of a linear
system into geometric terms: by what factors x 1 and x 2 must we dilate the sides
of the starting parallelogram so that it will fill the other one?
We can use this picture, and our geometric understanding of determinants,
to get a new formula for solving linear systems. Compare the sizes of these
shaded boxes.
( 6
8)
( 1
x 1 ·
3)
( 1
3)
( 2
1)
( 2
1)
( 2
1)