## MORE ON KERNELS AND IMAGES

**
In taking powers of a matrix, in general, the kernel stays
the same or gets larger. The image stays the same or gets smaller. **
Try this with A =

0 1

0 0

Note that A^2 = 0.

In general, ker(B) is contained in ker(AB) and
im(BA) is contained in im(B). Try these out with
B = A^k.

im(B) is contained in ker(A) if and only if AB = 0.

The intersection of two subspaces is always a subspace --
go back to the definition
and check that: (0) 0 is in both (1) if v, w are in both
the v+w is in both (2) if v is in both and c is a scalar
then cv is in both.

Think about the union of the coordinate axes
in the plane. It is not a subspace of
the plane -- closure under + fails.

The columns of a SQUARE MATRIX are independent if and only if
the rref is the identity. In general, when the columns are
independent you get what looks like an identity at the top
and then some rows of zeros. The columns cannot be independent
unless the number of rows is at least as large as the number
of columns.

E.g. for

1 2

3 6

the columns are dependent. The rref is

1 2

0 0

which has the same relation on its columns: the second is
twice the first in both cases. Passing to the rref typically
changes the column space but not the relations that hold among
the columns.

Finding a basis for a subspace amounts to finding the smallest
number of vectors that span (they are then indep.
or you can drop at least one). It is also amounts to finding
the largest number in the subspace that are indep. (These must
span: a vector outside the span but in the subspace will enable
you to enlarge the indep. set.) The smallest number that span
and the largest number that are indep. both give the DIMENSION
of the subspace.

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