Understanding Identity Matrix in Matrix Transformations
Explore how the Identity Matrix plays a crucial role in understanding various transformations on points within matrices. Learn how the Identity Matrix affects transformations and the significance of AI = IA = A. Discover examples and matrices that illustrate these concepts clearly.
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Presentation Transcript
Matrix Transformations Transform a matrix Recognise the Identity Matrix Understand that AI = IA = A Understand how the Identity Matrix effects transformations
A point P (x, y) can be transformed to an image point, P (x ,y ). We say that P is mapped to the image P .
Example 1 When the point (4,2) is transformed by a reflection in the When the point (4,2) is transformed by a reflection in the y- -axis, the image point is ( the image point is (- -4, 2 4, 2). ). (4,2) is mapped to ( (4,2) is mapped to (- -4,2) 4,2) axis, Example 2 When the point (3, When the point (3,- -1) is transformed by a rotation through 270 1) is transformed by a rotation through 270o o clockwise about the origin, the image point is (1,3 clockwise about the origin, the image point is (1,3). ). (3, (3,- -1) is mapped to (1,3) 1) is mapped to (1,3) These transformations can be defined by a matrix. For a These transformations can be defined by a matrix. For a transformation that transformation that maps (1, 0) to ( maps (1, 0) to (a, c a, c) and (0, 1) to ( ) and (0, 1) to (b, d b, d) ) the the transformation transformationmatrix matrix is is
To summarise: To summarise: Point P ( Point P (x, , y) and the image point ) and the image point P' ( P' (x', ', y') are connected by: ') are connected by:
Example: Example: To work out the image of the point ( To work out the image of the point (- -2, 5) for the transformation defined by the matrix: the transformation defined by the matrix: 2, 5) for First, write the coordinates (-2, 5) as a 2 x 1 matrix. Then multiply this vector by the transformation matrix. Write the 2 x 1 Write the 2 x 1 matrix as matrix as coordinates. coordinates. The image point is The image point is (11,9). (11,9).
4) & (x, , y) ) a) (3,2) ( a) (3,2) (- -1,5) (6,0) ( 1,5) (6,0) (- -3, 3,- -4) & ( No No transformation has occurred transformation has occurred. . b b) (3, ) (3,- -2) ( 2) (- -1, 1,- -5) (6,0) ( 5) (6,0) (- -3,4) & ( Reflection Reflection in the in the x- -axis 3,4) & (x, , - -y) ) axis. .
This matrix is called the This matrix is called the Identity Matrix, Identity Matrix, I. . When When multiplied by another multiplied by another matrix, matrix, A: : AI = = IA = = A Subsequently, when Subsequently, when I is used as a transformation matrix, no change occurs. transformation matrix, no change occurs. is used as a
