6. Linear transformations. Consider the function. f : R 2 R 2 which sends (x, y) (x, y)

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1 Consider the function 6 Linear transformations f : R 2 R 2 which sends (x, ) (, x) This is an example of a linear transformation Before we get into the definition of a linear transformation, let s investigate the properties of this map What happens to the point (1, 0)? It gets sent to (0, 1) What about (2, 0)? It gets sent to (0, 2) In fact an point on the x-axis gets sent to a point on the -axis How about points on the -axis? (0, 1) gets sent to ( 1, 0) (0, 2) gets sent to ( 2, 0) and so on Points on the -axis are sent to points on the x-axis What happens to the point (1, 1)? It gets sent to ( 1, 1) We guess that this function rotates the plane through an angle of π/2 anticlockwise The ke thing is that this map is represented b a matrix Let 1 0 The first column is the image of the vector (1, 0) and the second column is the image of the vector (0, 1) To appl A to (x, ) simpl multipl ( x =, 1 0 ) x as required What about the map f : R 2 R 2 which sends (x, ) (x, ) This map fixes (1, 0) and in fact it fixes the whole x-axis It sends (0, 1) to (0, 1) It sends the -axis to the -axis but it flips it upside-down (1, 1) gets sent to (1, 1) This map represents reflection in the x-axis The corresponding matrix is 1 0 We check that this is correct: ( ( 1 0 x x =, ) ) as required How about the matrix 1 0? 1

2 It sends (x, ) to ( 1 0 x = ) x This sends (1, 0) to ( 1, 0) and (0, 1) to (0, 1) This represents rotation through an angle of π or, equivalentl reflection in the origin The map f(x, ) = (2x, 2) represents a dilation b a factor of 2 and the map g(x, ) = (3x, 3) represents a dilation b a factor of 3 The corresponding matrices are ( If we take the matrix ) and B = then the corresponding map is f(x, ) = (2x, ) this magnifies distances in the x-direction b a factor of 2 and leaves heights unchanged How about the matrix 1 1? 0 1 It sends (x, ) to ( 1 1 x = 0 1) x + This sends (1, 0) to (1, 0) and (0, 1) to (1, 1) But it sends (1, 1) to (2, 1), (2, 1) to (3, 1) etc The higher up ou go, the more ou move horizontall This map is called a shear What about the function f : R 3 R 2 which sends (x,, z) (x, )? This represents projection onto the x-plane We just forget the height This sends (1, 0, 0) to (1, 0), (0, 1, 0) to (0, 1) and (0, 0, 1) to (0, 0) The corresponding matrix is We check that this is correct: x = z as required 2 ( x ),

3 Let The corresponding function is from R 3 to R 3, f : R 3 R 3 It sends (1, 0, 0) to (1, 0, 0), it sends (0, 1, 0) to (0, 1, 0) and it sends (0, 0, 1) to (0, 0, 1) Since A fixes (1, 0, 0), (0, 1, 0) and (0, 0, 1) it fixes everthing Let s check using matrix multiplication: x = x, z z so that f(x,, z) = (x,, z) is the identit function I 3 = is called the identit matrix The ke propert about linear transformations is one just needs to know what happens to (1, 0) and (0, 1), or more generall what happens to an collection of vectors which spans Definition 61 A function f : R n R m is called (a) linear (transformation) if (1) It is additive: f( v + w) = f( v) + f( w) for all vectors v and w R n (2) f(λ v) = λf( v), for all scalars λ and vectors v R n The second condition often turns up in engineering as: double the input, double the output Proposition 62 If f : R n R m is given b matrix multiplication, f( v) = A v, where A an m n matrix, then f is linear Proof f( v + w) = A( v + w) = A v + A w = f( v) + f( w) and f(λ v) = A(λ v) = λa( v) = λf( v) We have seen that a linear transformation is determined b its action on (1, 0) and (0, 1), etc 3

4 Definition 63 Let e 1 = (1, 0,, 0), e 2 = (0, 1,, 0) and e n = (0, 0,, 1) be the n unit coordinate vectors in R n In R 2 we have e 1 = (1, 0) and e 2 = (0, 1) and in R 3 we have e 1 = (1, 0, 0), e 2 = (0, 1, 0) and e 3 = (0, 0, 1) Note that e i has a 1 in the ith position and zeroes everwhere else The ke point is that: For example Similarl x = x 1 e 1 + x 2 e x n e n (x, ) = (x, 0) + (0, ) = x(1, 0) + (0, 1) = x e 1 + e 2 (x,, z) = (x, 0, 0)+(0,, 0)+(0, 0, z) = x(1, 0, 0)+(0, 1, 0)+z(0, 0, 1) = x e 1 + e 2 +z e 3 Theorem 64 If f : R n R m is linear then there is an m n matrix A such that f( v) = A v Proof Let v i = f( e i ) R m Let A be the matrix whose columns are the vectors v 1, v 2,, v n Then f( x) = f(x 1 e 1 + x 2 e x n e n ) Consider the matrix equation = f(x 1 e 1 ) + f(x 2 e 2 ) + + f(x n e n ) = x 1 f( e 1 ) + x 2 f( e 2 ) + + x n f( e n ) = x 1 v 1 + x 2 v x n v n = A x A x = b Suppose that no matter the choice of b we can alwas solve this equation, that is, this equation is alwas consistent In terms of functions, given an b R m we ma find x R n such that f( x) = b Definition 65 We sa that f : R n R m is onto if given an b R m we can find x R n such that f( x) = b All of the functions R 2 R 2 above are onto Projection from R 3 R 2 is onto The identit is onto The map R 2 R 2 given b (x, ) (x, 0) is not onto The vector (0, 1) is not in the image, that is, we cannot find a vector mapping to (0, 1) 4

5 There is another natural question we can ask about matrix equations: A x = b Suppose that no matter the choice of b there are never infinitel man solutions This is the same as saing there are never two or more solutions In terms of functions there are never two vectors x 1 and x 2 such that f( x 1 ) = f( x 2 ) = b Definition 66 We sa that f : R n R m is one to one if given an b R m there is at most one vector x such that f( x) = b All of the functions R 2 R 2 above are one to one The function f : R 2 R 3 given b (x, ) (x,, 0) is one to one Projection from R 3 R 2 is not one to one The vector (0, 0, 1) and the vector (0, 0, 2) are both sent to (0, 0) In fact Theorem 67 Let A be an n m matrix The function f : R n R m given b x A x is one to one if and onl if the homogeneous equation has onl the trivial solution x = 0 A x = 0 Proof f is one to one if and onl if A x = b has at most one solution, for an b If f is one to one then ever equation has at most one solution and so the homogeneous has onl the trivial solution Now suppose that the homogeneous has onl the trivial solution Then the solution to an consistent equation A x = b is of the form a particular solution plus an solution to the homogeneous As the homogeneous has onl one solution there is onl one solution to A x = b 5

Math 369 Exam #1 Practice Problems

Math 369 Exam #1 Practice Problems Math 69 Exam # Practice Problems Find the set of solutions of the following sstem of linear equations Show enough work to make our steps clear x + + z + 4w x 4z 6w x + 5 + 7z + w Answer: We solve b forming