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l6 - Math 20F Linear Algebra Lecture 6 1.8-1.9 Linear...

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Math 20F Linear Algebra Lecture 6: 1.8-1.9 Linear Transformations. A transformation (or mapping or function ) T : R n R m is a rule that for each x R n assigns a vector T ( x ) R m , called the image of x . Matrix multiplication by an m × n matrix A gives a mapping R n 3 x A x R m : x 1 . . . x n a 11 x 1 + a 12 x 2 + ... + a 1 n x n a 21 x 1 + a 22 x 2 + ... + a 2 n x n . . . a m 1 x 1 + a m 2 x 2 + ... + a mn x n . A matrix transformation T ( x ) = A x is the simplest type of transformation. Ex 1 x 1 x 2 0 - 1 1 0 ‚ • x 1 x 2 = - x 2 x 1 rotates vectors an angle π/ 2 counterclockwise. Ex 2 x 1 x 2 3 0 0 3 ‚ • x 1 x 2 = 3 x 1 3 x 2 scales vectors by a factor 3. Ex 3 x 1 x 2 1 0 0 0 ‚ • x 1 x 2 = x 1 0 projects vectors onto the x 1 axis. Ex 4 x 1 x 2 1 0 0 - 1 ‚ • x 1 x 2 = x 1 0 reflects vectors about the x 1 axis. Ex 5 x 1 x 2 1 0 0 1 ‚ • x 1 x 2 = x 1 x 2 is the identity map. Ex 6 x 1 x 2 1 3 0 1 ‚ • x 1 x 2 = x 1 + 3 x 2 x 2 is called shear If T : R n R m is a transformation then the set R n is called the domain of T and R m is called the codomain . The set of all images T ( x
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