ONE-TO-ONE LINEAR TRANSFORMATIONS

Linear Algebra Project

Objectives:

Use 'linalg' package of Maple

1. To find whether a linear transformation is one-to-one.

2. To find the inverse transformation whenever possible.

Required Information:

Then

This also means A is invertible.

If is one-to-one then the inverse transformation is defined by

Solved Example 1:

Let A =

The transformation

is given by = - 2 + 2

= 2 + +

= +

a) Determine whether is one-to-one.

b) If is one-to-one find

Solution:

a) First we check if A is invertible using the inverse command.

A =

> with(linalg):

> A := matrix([[1, -2, 2], [2, 1, 1], [1, 1, 0]]);

> Ainverse:=inverse(A);

b) The existence of the inverse of A shows that T is one-to-one, and the range of T is . In fact for any vector

w = in there is a pre-image given by w.

> w := matrix(3,1,[w1, w2, w3]);

> x= evalm(Ainverse&*(w));

The vector x is given by =

Solved Example 2:

Let A =

is given by = + 4 -

= 2 +7 +

= +3

a) Show that is not one-to-one.

b) Find all vectors x in which are mapped onto the 0 vector.

(Note that: For a one-to-one transformation only the 0- vector is mapped onto 0 vector.

In other words Ax = 0 or A = has only the trivial solution

or the solution space of the system of corresponding linear equations consists of = 0, = 0, = 0

Solution:

(a) We check whether matrix A = is invertible.

> A:= matrix(3,3, [1,2,1,-2,1,4,7,4,-5]);

> inverse(A);

Problem 1:

A =

Problem 2:

A =

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MSIP Grant #P120A80089-98: "Three Urban Calculus Reform programs: Adopting the Best" 1998-2001, MSEIP Grant #P120A010031: "Four Colleges: Calculus + Enhancements" 2001-04