This lesson should help you work on section 1.9 in the third edition or Section 1.8 in the second updated edition.

Transcription

1 Lesson : his lesson should help you work on section.9 in the third edition or Section.8 in the second updated edition. In the last section, we saw many examples of linear transformations with definition xaxwhereaisamatrixofappropriatesize. In this section, we shall see that for any linear transformation :R n R m, we can find a matrix A mn such that (x)ax. he construction of A is based on the proof of the theorem in section.9 in the third edition and Section.8 in the second updated edition. Here is an example: Let : R 3 R be a linear transformation. x any x x herefore x x x x x x For example, Let : R 3 R be defined by x x x x 7 x x 3 Note that,, 7 3

2 x x x x x x x x x x x x 7 3 x x 7 3 x x herefore if A (x)ax 7 3 We call such a matrix the standard matrix of the linear transformation. Notation: In R 3, we use the noations e, e, e 3 Note that these vectors are the columns of the 3x3 Identity matrix. Use similar notations for R n for any n. As another example, we may look at the exercise on the page 9 of the third edition, which gives us : R 3 R, defined by (e )(,3), (e )(,7), (e 3 )(5,). o find the standard matrix of. Just as above, the standard matrix is obtained by arranging the images of

3 e,e,e 3 as the columns of the matrix in order. i.e. the standard matrix Please read heorem and understand the proof and please make sure that you know all the geometric linear transformations of R that are illustrated in this section. Note that to define a linear transformation, :R n R m, it is enough to define it on the column vectors of I nn the nxn identity matrix. Before proceeding ahead, let us recall from Unit Circle rigonometry, that for a circle of unit radius and center O, the coordinates of a point P on the corcumference can be written in terms of the angle t that OP subtends with the x -axis. x P (rcost, rsint) O t x 3

4 Exercise on page 9 in the third edition: o obtain the standard matrix of the linear transformation :R R that rotates the points in the plane about the origin through. e, e, e e to obtain the standard matrix, it is enough to obtain (e ) and (e )

5 e π π cos,sin (e) π / - e π / (e) π π cos,sin - herefore cos e sin e cos e sin e herefore the standard matrix of this linear transformation is he exercise is done, but just to see how this works, take any vector in R say x (x) 3 3 5

6 Note that. 88. x P x - O x 3 m POQ = 5 - (x) Q 6. o write the standard matrix of :R R given that (e )e (e )e 3e. (e )e (e ) (e )e 3e (e ) 3 (e ) 3 3 herefore the standard matrix is A. Given a linear transformation :R R which first reflects points through the vertical 6

7 x -axis and then rotates the points radians. o find the standard matrix. First, let us look at (e ) First, let us look at the reflection of e by the x -axis. - reflection of e by x axis e Note that the reflection maps to. Now the rotation by will map it to as shown below 7

8 x - reflection of e by x axis x e Final destination of e under the action of Note that reflection of e clock wise it should land on herefore e by the x -axis is e itself, when rotated by counter e and the standard matrix is he question is answered, still let us verify this on a R. when reflected by the x axis goes to clock wise by it is mapped to as shown below. when rotated counter 8

9 a reflected by the x axis a ROAION BY 9 DEGREES and also 8. in section.9 the third edition Given that x,x x 3x,x x,,x Remember, to obtain the matrix of this linear transformation, all that we have to do is find and and show that the above mapping can be obtained by multiplication by A. 9

10 3 3 A 3 3x x x x x Ax x x herefore (x)ax and :R R is a linear transformation. Exercise # section.9 hird Edition: Given a linear transformation :R R 3 such that x,x x x,x 3x,3x x o find x such that x,x,,9 Solution: wemayrewritethemapas x x 3 3 x x o find x x x such that (x) 9

11 isthesameastofindx x x such that 3 3 x x 9 Look at the augmented matrix 3, row echelon form: herefore the answer is x 3 Check: 5,35 3,5 3 3,3 5 3,,9 Note: From this point on we shall put a strong emphasis on writing proofs. I shall write some proofs in the lessons. he best way to learn writing proofs is to read the text has very nicely written for you. Before you read the rest of the lesson, please read the definitions of an onto mapping and a one to one mapping on the page 87 in the third edition and page... in the second updated edition. Recall that a one-one transformation assigns to each element in the domain a unique element in the range. For example:

12 Is a one-one map. But is NO one-one. Algebraically: :R n R m is one-one if (u)(v) u v. Note that the theorems and (please make sure that you read and understand the proofs) in the section.9 the third edition or.8 in the second updated edition, state that A linear transformation :R n R m with matrix A is one-one iff (x)x or iff

13 he columns of A are linearly independent. o see an illustration, consider : R 3 R 3 defined by x x x x 5 5, row echelon form: 3 Note that the columns are linearly dependent therefore the linear transformation is not one-one. Furthermore, note that as well as 5 Showing that there are more than one element with as the image. Recall that a linear transformation :R n R m isontoiffforeachbinr m we can find x in R n such that (x)b. In view of the theorem in section.9 hird Edition or in the section.8 in the second updated edition If A is the standard matrix of then is onto iff the columns of A span R m or in other words in each row in A has a pivot position. From the text: 6 on page 9 in the third edition: 3

14 o check if :R 3 R given by 3, 7, 5 is a) one-one b) onto First note that the standard matrix A Since the columns of this matrix can not be linearly independent (why?), therefore is not one one. additional practice, find u and v in R 3 such that (u)(u) A 5 3 7, row echelon form: each row has a pivot position therefore is onto on page 9 in the third edition: o check if :R R with standard matrix A Since is one to one , row echelon form: 7 9 columns are not linearly independent, the transformation is not one-one.

15 on page 9 in the third edition: o check if :R 5 R 5 with standard matrix A is onto , row echelon form: Since the last row does not have any pivot position, it is not Onto. Additional exercise: Find b in R 5 such that there is no x in R 5 with (x)b 5