Starting with Two Matrices
Gilbert Strang, MIT
1.
INTRODUCTION
The first sections of this paper represent an imaginary lecture,
very near the beginning of a linear algebra course. We chose two matrices
A
and
C
, on the
principle that examples are amazingly powerful. The reader is requested to be exceptionally
patient, suspending all prior experience—and suspending also any hunger for precision and
proof. Please allow a partial understanding to be established first.
I believe there is value in
naming
these matrices. The words “difference matrix” and
“sum matrix” tell how they act. It is the action of matrices, when we form
Ax
and
Cx
and
Sb
, that makes linear algebra such a dynamic and beautiful subject.
2.
A FIRST EXAMPLE
In the future I will begin my linear algebra class with these
three vectors
a
1
,
a
2
,
a
3
:
a
1
D
2
4
1
1
0
3
5
a
2
D
2
4
0
1
1
3
5
a
3
D
2
4
0
0
1
3
5
:
The fundamental operation on vectors is to take
linear combinations
.
Multiply these
vectors
a
1
,
a
2
,
a
3
by numbers
x
1
,
x
2
,
x
3
and add. This produces the linear combination
x
1
a
1
C
x
2
a
2
C
x
3
a
3
D
b
:
x
1
2
4
1
1
0
3
5
C
x
2
2
4
0
1
1
3
5
C
x
3
2
4
0
0
1
3
5
D
2
4
x
1
x
2
x
1
x
3
x
2
3
5
D
2
4
b
1
b
2
b
3
3
5
:
(1)
(I am omitting words that would be spoken while writing that vector equation.) A key step
is to rewrite (1) as a matrix equation:
Put the vectors
a
1
,
a
2
,
a
3
into the columns of a matrix
A
Put the multipliers
x
1
,
x
2
,
x
3
into a vector
x
A
D
2
4
a
1
a
2
a
3
3
5
D
2
4
1
0
0
1
1
0
0
1
1
3
5
x
D
2
4
x
1
x
2
x
3
3
5
Then
A
times
x
is exactly
x
1
a
1
C
x
2
a
2
C
x
3
a
3
. This is more than a definition of
Ax
, because
the rewriting brings a crucial change in viewpoint. At first, the
x
’s were multiplying the
a
’s.
Now, the matrix
A
is multiplying the vector
x
.
The matrix acts on
x
, to give a
combination of the columns of
A
:
Ax
D
2
4
1
0
0
1
1
0
0
1
1
3
5
2
4
x
1
x
2
x
3
3
5
D
2
4
x
1
x
2
x
1
x
3
x
2
3
5
D
2
4
b
1
b
2
b
3
3
5
:
(2)
When the
x
’s are known, the matrix
A
takes their differences.
We could imagine an
unwritten
x
0
D
0
, and put in
x
1
x
0
to complete the pattern.
A
is a
difference matrix
.
1
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One more step to a new viewpoint.
Suppose the
x
’s are not known but the
b
’s are
known. Then
Ax
D
b
becomes an equation for
x
, not an equation for
b
. We start with the
differences (the
b
’s) and ask which
x
’s have those differences.
Linear algebra is always interested first in
b
D
0
:
Ax
D
0
is
2
4
x
1
x
2
x
1
x
3
x
2
3
5
D
2
4
0
0
0
3
5
:
Then
x
1
D
0
x
2
D
0
x
3
D
0
(3)
For this matrix, the only solution to
Ax
D
0
is
x
D
0
. That may seem automatic but it’s not.
A key word in linear algebra (we are foreshadowing its importance) describes this situation.
These column vectors
a
1
,
a
2
,
a
3
are
independent
. The combination
x
1
a
1
C
x
2
a
2
C
x
3
a
3
is
Ax
D
0
only when all the
x
’s are zero.
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 Spring '08
 CHANILLO
 Linear Algebra, Algebra, Matrices

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