# Determine the kernel and range. HELP???

##
Determine the kernel and range of the transformation defined by the matrix

6 4

3 2

.

(Enter your answers as a comma-separated list. Enter each vector in the form

(x1, x2, ...).

Use r for any arbitrary scalar.)

Show that dim ker(T) + dim range(T) = dim domain(T).

dim ker(T) + dim range(T) = dim domain(T) right double arrow implies

Determine the kernel and range of the transformation defined by the matrix

6 4

3 2

.

(Enter your answers as a comma-separated list. Enter each vector in the form

(x1, x2, ...).

Use r for any arbitrary scalar.)

Show that dim ker(T) + dim range(T) = dim domain(T).

dim ker(T) + dim range(T) = dim domain(T) right double arrow implies

##### 1 Answer

#### Explanation:

The transformation takes the vector

**The range**

It is easy to see that

It is easy to see that all the elements of

**The Kernel**

The kernel of the transformation is defined by

So, if

Which will be satisfied by

It is obvious that

**Rank-nullity theorem**

Since

Thus the rank-nullity theorem

is verified by