Linear Transformations and matrices
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Matrix Transformations
The [Equation goes here - download the original to see it.] square matrix In the matrix [Equation goes here - download the original to see it.] the vectors [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] represent the images of [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] under M respectively. Example Find the image of [Equation goes here - download the original to see it.] and v under the matrix [Equation goes here - download the original to see it.] Solution This simple follows from matrix multiplication [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] You can see that this argument does not depend on the specific form of the matrix. That is, [Equation goes here - download the original to see it.] is the image of [Equation goes here - download the original to see it.] under [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] is the image of [Equation goes here - download the original to see it.] under [Equation goes here - download the original to see it.].Thus any [Equation goes here - download the original to see it.] matrix transforms a parallelogram, OXYZ, with one vertex at the origin to another parallelogram. The position of the origin is fixed - that is, it is not moved. Example Sketch the image of the square with vertices at the origin, and at [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] under the matrix [Equation goes here - download the original to see it.] Solution We have already seen that the image of [Equation goes here - download the original to see it.] under M is and the image of [Equation goes here - download the original to see it] under M is [Equation goes here - download the original to see it.]. We can find the image of [Equation goes here - download the original to see it.] as [Equation goes here - download the original to see it.] This gives sketch [Diagram goes here - download the original to see it.] We can see once again that the image forms a parallelogram regardless of what numbers are substituted in the matrix [Equation goes here - download the original to see it.] . This is because the image of [Equation goes here - download the original to see it.] under this matrix is [Equation goes here - download the original to see it.] So the fourth corner of the parallelogram is always the vector sum of the coordinates of the other two corners (assuming the parallelogram has one corner at the origin). However, we must include the case of the "degenerate" parallelogram - that is, the case were the parallelogram OXYZ collapses into a line under the matrix transformation. This occurs with the matrix is not singular. Example (a)Sketch the image of the square with vertices at the origin, and at [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] under the matrix [Equation goes here - download the original to see it.] .(b)What is the determinant of M? Solution (a)[Equation goes here - download the original to see it.] , [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] he points [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] all lie on the same line [Equation goes here - download the original to see it.]. [Diagram goes here - download the original to see it.] (b) [Equation goes here - download the original to see it.] So M is a singular matrix. We know that every singular matrix has row (and column) vectors that are linear combinations of each other, so if M is singular, then the image of [Equation goes here - download the original to see it.] under M is a linear combination of the images of [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] . This means that the image of a parallelogram with one vertex at the origin of a singular matrix is a line. Conversely, if the image of a parallelogram collapses into a line under a matrix M then that matrix must be singular.
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Contents of
Linear Transformations and matrices
1 Matrix Transformations
2 Enlargements, shears and rotations
3 Reflections
4 Combinations of rotations and reflections
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