MATH 280, Spring 2002 MATH 280, Spring 2002

Theme II: Linear Transformations



The due date for this theme is Friday, February 8, at noon.

Reading assignment: Section 1.8

Background reading: Section 1.7, 1.9



Let T be a transformation mapping Rn to Rm. We say that T is linear if it satisfies two properties:

  1. T( u+v) = T( u) +T( v) for all u and v in Rn;

  2. T( cu) = cT( u) for all u in Rn and all scalars c.

Repeated application of these two properties yields the important superposition principle for linear transformations:

T( c1v1+¼+cvp) = c1T( v1) +¼+cpT( vp) ,
in which v1,¼,vp are vectors and c1,¼,cp are scalars. Our first goal of this theme is to use superposition to justify that a linear transformation is completely determined by its action on the column vectors ej of the identity matrix.

Suppose that

x = é
ê
ê
ê
ê
ë
x1
x2
x3
 ù
ú
ú
ú
ú
û
is a vector in R3. Explain how you can write x as a linear combination of the columns e1,e2,e3 of the identity matrix on R3, I3. Now, let T be a linear transformation the sends vectors of R3 to vectors in Rn. Use your previous result and superposition to explain why we know T( x) , the image of x under T, once we know T( e1) , T( e2)  and T( e3) .

Superposition allows us to see how linear transformations preserve properties of sets of vectors. One such property is linear dependence. Suppose that T: Rn® Rm is a linear transformation and { v1,v2,v3} are linearly dependent vectors in Rn. Prove that {T( v1) ,T( v2) ,T( v3) } are linearly dependent vectors in Rm.

It may be somewhat surprising that a linear transformation does not preserve linear independence. Give an example of a linear transformation T: R2® R2 and a pair of linearly independent vectors v1 and v2 in R2 such that T( v1)  and T( v2) are not linearly independent. Be sure to show that T is linear, v1 and v2 are linearly independent, and T( v1)  and T( v2) are not linearly independent.

If T: Rn® Rm is a linear transformation, then there is a matrix A that represents T; that is

T( x) = Ax
for all x in Rn. This matrix is called the standard matrix for the linear transformation T. If we know the image of ej under T for all j, it is easy to write the standard matrix. However, we may still be able to find the standard matrix if we know the action of T on enough vectors. Suppose that
T æ
ç
è 		é
ê
ë
1
2
 ù
ú
û 		ö
÷
ø 		= é
ê
ë
2
-1
 ù
ú
û 		and T æ
ç
è 		é
ê
ë
1
1
 ù
ú
û 		ö
÷
ø 		= é
ê
ë
2
1
 ù
ú
û
Find the standard matrix for T and explain your work.

Many problems related to the use of computers involve linear systems. In the following problem, we will use the fact that the composition of linear transformations is a linear transformation. Before solving the problem below, prove this result. That is, suppose that T: Rn® Rm and S: Rk® Rn are linear transformations. Prove that the composition T°S defined by (T°S) ( x) = T( S( x)) is a linear transformation.

We are now ready to solve the following problem:

Modern printers have the ability to print in two different orientations: portrait (the standard orientation of a sheet of letter-sized paper which is taller than it is wide) and landscape (the paper is rotated so that it is wider than it is tall). To save paper when printing a long document, it is often convenient to print two pages of text side-by-side in landscape mode. To do this, one imagines two sheets of letter sized paper placed adjacent to each other. The pages are then rotated and either contracted or dilated in both length and width to fit on a single letter-sized sheet.

Find three linear transformations which perform these operations. One will rotate the pages, another will adjust the height, and the third will adjust the width. Identify which matrix accomplishes which task and explain why each does so.

File translated from TEX by TTH, version 1.95.
On 28 Jan 2002, 14:17.