Homework due Class 25

Find the kernel and range of the following linear transformations:

1. The transformation T from C[0,1] to R which takes every function f(x) to the number f(1). The kernel is defined as the set of all vectors A in V such that f(A)=0, that is

ker(T)={A in V | T(A)=0}

The kernel of the transformation in the first problem is the set of all functions f(x) such that f(1)=0.

The range is defined as the set of all vectors in W which are images of some vectors in V, that is

range(T)={A in W | there exists B in V such that T(B)=A}.

The range of the transformation in the first problem is the set of all numbers, that is range(T)=R. Indeed, take any number A. Take the constant function f_A(x)=A. Then f_A belongs to C[0,1] and T(f)=f(1)=A. So A is in the range of T. Since A was an arbitrary number from R, the proof is complete.

2. The operator on the space of 2 by 2 matrices to itself which takes every matrix

[ a	b ]
[ c	d ]

to the product

[ 1 2 ] [ 0 1 ]

[ a b ] [ c d ]

(Hint: use the fact that the matrix

[ 1	2 ]
[ 0	1 ]

is invertible.).

The kernal of this consists only of the zero matrix, since any invertible matrix multiplied by a non-zero matrix will have non-zero result.

The range of this transformation consists of all 2 by 2 matrices. Indeed, let A be a 2 by 2 matrix. In order to prove that A is in the range of our transformation, we need to find a 2 by 2 matrix B such that

A

=

[ 1 2 ]

*

B

[ 0

1 ]

Since

[ 1	2 ]
[ 0	1 ]

is an invertible matrix, we can get B by multiplying A on the left by the inverse of this matrix. Thus B exists and A is in the range of our linear operator. Since A was an arbitrary 2 by 2 matrix, the proof is complete.