## FINDING A BASIS FOR THE KERNEL OR IMAGE

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To find the kernel of a matrix A is the same as to solve the
system AX = 0, and one usually does this by putting A in rref.
The matrix A and its rref B have exactly the same kernel. In both
cases, the kernel is the set of solutions of the corresponding homogeneous
linear equations, AX = 0 or BX = 0.
**
You can express the solution set as a linear combination of certain
constant vectors in which the coefficients are the free variables.

E.g., to get the kernel of

1 2 3

2 4 6

one gets the rref

1 2 3

0 0 0

and then one solves x+2y+3z = 0 (this is already reduced). The general
solution is

-2y-3z

{ y : y, z in R}

z

which you can write as

-2 -3

y 1 + z 0

0 1

The column vectors

-2

1

0

and

-3

0

1

span the kernel, clearly. They are independent because, each one,
in the coordinate spot corresponding to the free variable which
is its coefficient, has a 1, while the other vector(s) have a 0
in that spot.

So the vectors produced to span the kernel by this method are
always a basis for the kernel, and the dimension of the kernel
= number of free variables in solving AX = 0.

In getting a basis for the image one wants to pick out certain
columns. The relations on the columns of the rref are the same
as the relations on the columns of the original matrix. (Solutions
of the equations again.) Therefore, if a set of columns of the
rref is a basis for the image of the rref, the CORRESPONDING columns of
the original matrix A are also a basis. One thing that always works
is to use the pivot columns of the original matrix: these are
the columns where the rref has leading ones.

For example, consider

0 0 0

1 2 3

2 4 7

The rref is

1 2 0

0 0 1

0 0 0

The pivot columns are the first and third.
This shows that the first and third columns of the original
matrix are a basis for its image. HOWEVER, these two matrices
do not have the same image.

The simplest example where a matrix A and its rref do not have
the same image (column space) is when A =

0

1

The column space is the line spanned by that vector: the e_2 or y-axis.

But the rref is

1

0

and the column space is the line spanned by that one vector: the
e_1 or x-axis.

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