Dr. Byrne Math 237Fall 2009 Section 1Worksheet on Linear TransformationsLet A be an m x n matrix. Then Ax (for x  n) yields a linear transformation TA: n  m. The matrix A represents the transformation TA and the transformation TA is said to be induced by A. We will look at some special cases of linear transformations.I. The Identity Transformation TIThe identity transformation is the transformation TI: n  n that maps every vector x in n to itself. This transformation is induced by thenn  identity matrix. A. The Identity Transformation TI: 2  2The matrix representing T1 I: 2  2 is1001A. Find the image of 107xunder this transformation by finding Ax:107T= Ax =1001107=B. The Identity Transformation TI: 3  3Write down the matrix that represents the identity transformation for vectors in 3.II. The Zero Transformation T0The zero transformation is the transformation T0: n  m that maps every vector x in n to the zero vector in m (i.e., T(x)= 0). This transformation is induced by the mxn zero matrix. A. The Zero Transformation T0: 3  2The matrix representing T0 is000000A. Find the image of 11107xunder this transformation by finding Ax:11107T= Ax =00000011107=B. The Zero Transformation T0: 2  7Which of these matrices has the correct dimensions for inducing T0 (x) = Ax? Hint: The transformation needs to transform a vector x in 2 to the zero vector in 7 so you can check by trying to apply the transformation to a vector in 2 (by computing Ax). 00000000000000 or 00000000000000?C. The Zero Transformation T0: 3  1Write down the matrix T0 so that Ax will transform x from a vector in 3 to a vector in 1.III. Finding the matrix A that induces TA:  n   mGiven a particular linear transformation T: n  m, it is easy to find a matrix A that represents T. Start with the standard basis vectors for n:{e1, e2, …, en}. Under the transformation T, suppose these vectors map to {T(e1), T(e2), …, T(en)}. Then the matrix that represents T is given by [T(e1), T(e2), …, T(en)]. A. Example: Recall that the zero transformation T0: 3  2 is induced by000000A. The standard basis vectors of 3 are 100,010,001321eee and they are transformed into the zero vector in 2:00)(,00)(,00)(321eTeTeT. Thus the matrix that represents T0 is given by )()()(321eTeTeT: 000000)()()(321eTeTeTAB. Find the matrix that represents a transformation Find the matrix A that induces the transformation 322121xxxxxxT.01T10T1001TTATFind 34Tusing the rule 322121xxxxxxT:Find 34Tby calculating AT x:IV. Reflections of the form T :  2   2The basis vectors for 2 are 01 and10.A. A reflection about the x-axis will take a point (x,y) to (x,-y). Plot the basis vectors of 2 and their image after reflection about the x-axis on the same graph. Write down thematrix A that represents this transformation and use Ax to calculate the image of34x under this transformation.B. A reflection about the y-axis will take a point (x,y) to (-x,y). Plot the basis vectors of 2 and their image after reflection about the y-axis on the same graph. Write down the matrix A that represents this transformation and use Ax to calculate the image of34x under this transformation.C. A reflection about the line y=x will take a point (x,y) to (y,x). Plot the basis vectors of 2 and their image after reflection about the y-axis on the same graph. Write down the matrix A that represents this transformation and use Ax to calculate the image of34x under this transformation.V. Contractions and Dilations in  2In 2, contraction and dilation are represented by cc00where c is a scalar. If c<1 then the transformation is a contraction, if c>1, the transformation is a dilation. VI. Rotations in  2To rotate a vector counter-clockwise by radians, we can use the transformation induced by T= cossinsincos.A. Find the matrix which represents a transformation that rotates vectors by /2 radians. Find the images of01, 10 and 34. Plot these three vectors with their images. B. Find the matrix which represents a transformation that rotates vectors by /4 radians. Find the images of01, 10 and 34. Plot these three vectors with their images. C. Find the matrix which represents a transformation that rotates vectors by 0 radians. Find the images of 01, 10 and 34. Plot these three vectors with their images. VII. Composition of TransformationsSuppose that a transformation TA maps a vector from n to a vector in m in and that a transformation TB maps a vector from m to a vector in k. Then the composition of TA with TB is denoted by TB  TA or TB (TA (x)). First the transformation TA is applied, and then the transformation TB is applied and the composition transformation will take a vector in n to a vector in k. (I.e., we have nmk.) If TA is induced by A (TA=Ax) and TB is induced by B (TB=Bx), then the composite transformation TB  TA is given by BA (by