Linear Transformations - CHAPTER 6 LINEAR TRANSFORMATIONS...

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Unformatted text preview: CHAPTER 6 LINEAR TRANSFORMATIONS 6.1 WHAT IS A LINEAR TRANSFORMA- TION? You know what a function is - it’s a RULE which turns NUMBERS INTO OTHER NUMBERS: f ( x ) = x 2 means “please turn 3 into 9, 12 into 144 and so on”. Similarly a TRANSFORMATION is a rule which turns VECTORS into other VECTORS. For exam- ple, “please rotate all 3-dimensional vectors through an angle of 90 ◦ clockwise around the z-axis”. A LIN- EAR TRANSFORMATION T is one that ALSO satisfies these rules: if c is any scalar, and * u and * v are vectors, then T ( c * u ) = cT ( * u ) and T ( * u + * v ) = T ( * u ) + T ( * v ) . 1 EXAMPLE : Let I be the rule I * u = * u for all * u . You can check that I is linear! Called IDENTITY Linear Transformation. EXAMPLE : Let D be the rule D * u = 2 * u for all * u . D ( c * u ) = 2( c * u ) = c (2 * u ) = cD * u D ( * u + * v ) = 2( * u + * v ) = 2 * u + 2 * v = D * u + D * v → LINEAR! Note: Usually we write D ( * u ) as just D * u . 6.2. THE BASIC BOX, AND THE MATRIX OF A LINEAR TRANSFORMATION The usual vectors ˆ i and ˆ j define a square: Let’s call this the BASIC BOX in two dimensions. 2 Similarly, ˆ i, ˆ j , and ˆ k define the BASIC BOX in 3 dimensions. Now let T be any linear transformation. You know that any 2-dimensional vector can be written as a ˆ i + b ˆ j , for some numbers a and b . So for any vector, we have T ( a ˆ i + b ˆ j ) = aT ˆ i + bT ˆ j. This formula tells us something very important: IF I KNOW WHAT T DOES TO ˆ i and ˆ j , THEN I KNOW EVERYTHING ABOUT T- because now I can tell you what T does to ANY vector. EXAMPLE : Suppose I know that T ( ˆ i ) = ˆ i + 1 4 ˆ j and T ( ˆ j ) = 1 4 ˆ i + ˆ j . Then what is T (2 ˆ i + 3 ˆ j )? Answer : T (2 ˆ i + 3 ˆ j ) = 2 T ˆ i + 3 T ˆ j = 2 ‡ ˆ i + 1 4 ˆ j · + 3 ‡ 1 4 ˆ i + ˆ j · = 2 ˆ i + 1 2 ˆ j + 3 4 ˆ i + 3 ˆ j = 11 4 ˆ i + 7 2 ˆ j . Since T ˆ i and T ˆ j tell me everything I need to know, 3 this means that I can tell you everything about T by telling you WHAT IT DOES TO THE BASIC BOX. EXAMPLE : Let T be the same transformation as above, T ( ˆ i ) = ˆ i + 1 4 ˆ j and T ( ˆ j ) = 1 4 ˆ i + ˆ j . The basic box has been squashed a bit! Pictures of WHAT T DOES TO THE BASIC BOX tell us everything about T ! EXAMPLE : If D is the transformation D * u = 2 * u , then the Basic Box just gets stretched: So every LT can be pictured by seeing what it does to the Basic Box. There is another way! Let T ˆ i = • a c ‚ and T ˆ j = • b d ‚ . Then we DEFINE 4 THE MATRIX OF T RELATIVE TO ˆ i, ˆ j as • a b c d ‚ , that is, the first COLUMN tells us what happened to ˆ i , and the second column tells us what happened to ˆ j ....
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Linear Transformations - CHAPTER 6 LINEAR TRANSFORMATIONS...

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