1. Working in R2, a. Find the standard matrix for a transformation T that stretc

Question

1. Working in R2, a. Find the standard matrix for a transformation T that stretches the unit square 2 in the horizontal direction and 3 in the vertical direction. b. Find the standard matrix for a transformation Q that rotates the unit square 60 degrees in the counterclockwise direction. c. Find a matrix for the composed transformation T(Q(x)) using matrix multiplication. Is this equivalent to the composed transformation Q(T(x))? d. Is the matrix you found in the last part through matrix multiplication the standard matrix for the combined transformations? Explain why or why not.

Explanation / Answer

follow this In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy-Cartesian plane counterclockwise through an angle ? about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system. Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In two-dimensional space, a rotation can be simply described by an angle ? of rotation, but it can also be represented by the four entries of a rotation matrix with two rows and two columns. In three-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem), and hence it can be simply described by an angle and a vector with three entries. However, it can also be represented by the nine entries of a rotation matrix with three rows and three columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement, which can be represented by a matrix, but not associated single axis or angle. Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1: . The set of all such matrices of size n forms a (generally not commutative) group, known as the special orthogonal group SO(n). In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant -1 (instead of +1). These combine proper rotations with reflections (which invert orientation). In this sense the members of the (general) orthogonal group O(n) may also be called rotation matrices. In other cases, where reflections are not being considered, the label proper may be dropped unless demands of clarity make it wise to specify it. This is the reason it is more frequent to associate the term rotation matrix with members of SO(n) (as in the rest of this article) while it may also be found referring to members of O(n). Contents [show] [edit]In two dimensions A counterclockwise rotation of a vector through angle ?. The vector is initially aligned with the x-axis. In two dimensions every rotation matrix has the following form: . This rotates column vectors by means of the following matrix multiplication: . So the coordinates (x',y') of the point (x,y) after rotation are: , . The direction of vector rotation is counterclockwise if ? is positive (e.g. 90°), and clockwise if ? is negative (e.g. -90°). . Note that the two-dimensional case is the only non-trivial (e.g. one dimension) case where the rotation matrices group is commutative, so that it does not matter the order in which multiple rotations are performed. [edit]Non-standard orientation of the coordinate system A rotation through angle ? with non-standard axes. If a standard right-handed Cartesian coordinate system is used, with the x axis to the right and the y axis up, the rotation R(?) is counterclockwise. If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R(?) is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the y-axis down the screen or page.[1] See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix. [edit]Common rotations Particularly useful are the matrices for 90° and 180° rotations: (90° counterclockwise rotation) (180° rotation in either direction – a half-turn) (270° counterclockwise rotation, the same as a 90° clockwise rotation) [edit]In three dimensions See also: Rotation formalisms in three dimensions [edit]Basic rotations The following three basic (gimbal-like) rotation matrices rotate vectors about the x, y, or z axis, in three dimensions: Each of these basic vector rotations appears counter-clockwise when the axis about which they occur points toward the observer, and the coordinate system is right-handed. Rz, for instance, would rotate toward the y-axis a vector aligned with the x-axis. This is similar to the rotation produced by the above mentioned 2-D rotation matrix. See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices. [edit]General rotations Other rotation matrices can be obtained from these three using matrix multiplication. For example, the product represents a rotation whose yaw, pitch, and roll are a, ß, and ?, respectively. Similarly, the product represents a rotation whose Euler angles are a, ß, and ? (using the y-x-z convention for Euler angles). The rotation matrix corresponding to a rotation with Euler angles , with x-y-z convention, is given by: These matrices produce the desired effect only if they are used to pre-multiply column vectors (see Ambiguities for more details). [edit]Conversion from and to axis-angle Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis (Euler rotation theorem). There are several methods to compute an axis and an angle from a rotation matrix (see also axis-angle). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix. [edit]Determining the axis A rotation R around axis u can be decomposed using 3 endomorphisms P, (I - P), and Q (click to enlarge). Given a rotation matrix R, a vector u parallel to the rotation axis must satisfy since the rotation of around the rotation axis must result in . The equation above may be solved for which is unique up to a scalar factor. Further, the equation may be rewritten which shows that is the null space of . Viewed another way, is an eigenvector of R corresponding to the eigenvalue (every rotation matrix must have this eigenvalue). [edit]Determining the angle To find the angle of a rotation, once the axis of the rotation is known, select a vector perpendicular to the axis. Then the angle of the rotation is the angle between and . A much easier method, however, is to calculate the trace (i.e. the sum of the diagonal elements of the rotation matrix) which is . [edit]Rotation matrix from axis and angle For some applications, it is helpful to be able to make a rotation with a given axis. Given a unit vector u = (ux, uy, uz), where ux2 + uy2 + uz2 = 1, the matrix for a rotation by an angle of ? about an axis in the direction of u is {{[2] }} This can be written more concisely as where is the cross product matrix of u, ? is the tensor product and I is the Identity matrix. This is a matrix form of Rodrigues' rotation formula, with If the 3D space is right-handed, this rotation will be counterclockwise for an observer placed so that the axis u goes in his direction (Right-hand rule). [edit]Properties of a rotation matrix

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