The matrix M =
[−3/5 4/5]
[4/5 3/5]
defines an isometry of the xy-plane.
(a)What special properties do the column vectors of this matrix have?
(b)Verify that the point (2, 4) remains stationary when M is applied to it.
(c)What is the significance of the stationary point (2, 4)? What does it tell you about the possible isometries that M could be? Do other points invite examination?
(d)Show that MM is the 2×2 identity matrix. What does this suggest about the geometric transformation that M represents? Confirm your suspicions.
To answer these questions, let's go step by step:
(a) The special properties of the column vectors of matrix M can be determined by examining their magnitudes. For any matrix representing an isometry, the column vectors should have unit magnitudes (lengths of 1). Let's calculate the magnitudes of the column vectors:
Magnitude of the first column vector = sqrt((-3/5)^2 + (4/5)^2) = sqrt(9/25 + 16/25) = sqrt(25/25) = 1
Magnitude of the second column vector = sqrt((4/5)^2 + (3/5)^2) = sqrt(16/25 + 9/25) = sqrt(25/25) = 1
As both column vectors have magnitudes of 1, they satisfy the special property required for an isometry.
(b) To verify if the point (2, 4) remains stationary when M is applied, we need to multiply the matrix M with the coordinate vector of the point (2, 4) and see if the result is equal to the original vector.
Matrix M multiplied by the point (2, 4):
[(-3/5)(2) + (4/5)(4)] = (0)
[(4/5)(2) + (3/5)(4)] = (4)
The resulting vector is (0, 4), which is not equal to the original vector (2, 4). Therefore, the point (2, 4) does not remain stationary under the transformation of matrix M. It is important to note that an isometry should keep all points stationary, so M may not represent an isometry.
(c) The significance of the stationary point (2, 4) is that it represents a fixed point that does not move under the transformation. This point can help determine the nature of the transformation represented by the matrix M. In this case, since the point does not remain stationary, it suggests that M does not represent a pure rigid motion isometry. Other points may also invite examination to verify their behavior under the transformation and provide further insight into the nature of M.
(d) To show that MM is the 2x2 identity matrix, we need to multiply the matrix M with itself:
[(-3/5)(-3/5) + (4/5)(4/5)] [(-3/5)(4/5) + (4/5)(3/5)]
[(4/5)(-3/5) + (3/5)(4/5)] [(4/5)(4/5) + (3/5)(3/5)]
Simplifying each entry:
[(9/25) + (16/25)] [(-12/25) + (12/25)]
[(-12/25) + (12/25)] [(16/25) + (9/25)]
This simplifies to:
[1] [0]
[0] [1]
As the resulting matrix is the 2x2 identity matrix, it suggests that the geometric transformation represented by M is the identity transformation. The identity transformation keeps all points stationary, meaning that M does not produce any change to the position or orientation of vectors.
By answering these questions, we have gained a deeper understanding of the properties and behavior of the given matrix M.