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Solution Set Bretscher 2.1 - 3,4,24-30,38,42,43,44 6/28/17 2.1 / 3 x1 x2 x3 In order for a transformation T to be linear, T(x+y) = T(x) + T(y). For the given transformation T x2 x1 x3 , we x x x 3 1 2 1 1 2 0 would expect T 1 1 T 2 4 1 1 2 0 transformation is not linear. ? 1 1 0 0 0 T 1 T 1 1 1 2 . Clearly, this is not the case. Thus, this 1 1 0 0 0 2.1 / 4 y1 y2 We can rewrite the transformation y3 y4 9 x1 2 x1 4 x1 5 x1 3 x2 9 x2 9 x2 x2 3x3 x3 in matrix notation as 2 x3 5 x3 y1 9 3 3 y2 2 9 1 x1 y 4 9 2 x2 . Thus, the 3 5 1 5 x3 y4 4x3 matrix gives the matrix of the linear transformation. 2.1 / 24 This is a rotation transformation by 90° counterclockwise. 2.1 / 27 This is a reflection about the x-axis. 2.1 / 30 This is a projection onto the y-axis. (Drawing should be completely flat.) 2.1 / 28 This is a scaling in the y direction by 2. 2.1 / 25 This is a scaling by 2. 2.1 / 29 This is a reflection across both the x and yaxis, or equivalently, a rotation by 180°. 2.1 / 26 This is reflection across y=x. 2.1 / 38 Since v1 and v2 are the column vectors, we know that T e1 v1 and T e2 v2 . Thus, T 2 T 2e1 e2 2T e1 T v2 2v1 v2 . 1 2.1 / 42 0 a. T 0 0 0 0 1 1 T 1 21 0 2 b. 2v1 v2 v2 1 1 T 0 12 0 2 0 T 1 1 0 0 0 T 0 0 1 1 0 T 1 1 1 1 1 1 T 0 12 1 2 1 1 T 1 12 1 2 v1 x3 x2 x1 The points that are transformed to 0 satisfy the following system of equations: 0 x1 2t 1 1 0 0 1 2 0 0 1 0 2 0 12 x1 x2 0 12 0 1 1 0 0 1 1 0 x2 t 1 2 x1 x3 0 x t 2 0 1 0 3 2.1 / 43 a. We can show that transformation T is linear by finding a matrix of T. Recalling the definition of matrix multiplication, the formula for T is similar to the matrix multiplication of a row vector by a column vector. In fact, the row vector is basically v expressed as a row vector. Thus, the matrix of T is 2 3 4 . b. More generally, the matrix of T for any vector v is v1 v2 v3 , or v t (the transpose of v ). c. Conversely, given any linear transformation T from 3 to , we know that the corresponding matrix must be a 1x3 matrix v1 v2 v3 . From part b, we can immediately see that this linear transformation is the corresponding matrix of v 1 the dot product transformation with v v2 . v 3 2.1 / 44 We can show that the cross product transformation is linear by finding the corresponding matrix directly. Alternatively, we can show that the cross product satisfies the two conditions for linearity: 1. T kx kT x T kx v kx k v x kT x 2. T x y T x T y T x y v x y v x v y T x T y In proving the above two conditions, we have used a few properties of cross product. Since both conditions are satisfied, the cross product transformation is linear. Using the definition of cross product, we can write: v x v x 0 v3 v2 x1 v x 1 1 2 3 3 2 T x v x v2 x2 v3 x1 v1 x3 v3 0 v1 x2 . Thus, we have found the transformation matrix. v x v x v x v 0 x3 3 3 1 2 2 1 2 v1