Suppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is reflection over the line y=−5/3x.
Ah, reflections over lines, a classic transformation! So, we want to find the matrix A that induces T, which is a reflection over the line y = -5/3x.
Well, let's break it down. First, a reflection over the x-axis can be represented by the matrix:
A1 = [[1, 0], [0, -1]]
And a reflection over the y-axis can be represented by the matrix:
A2 = [[-1, 0], [0, 1]]
Now, if we want to reflect over the line y = -5/3x, we can think of it as a combination of two reflections. We can first reflect over the y-axis, then reflect over the line y = 5/3x, and finally, reflect over the y-axis again. So, the matrix A that induces T would be:
A = A2 * A3 * A2
Now, the trick is to find A3, which represents the reflection over the line y = 5/3x. We can visualize it as reflecting over the y-axis, then reflecting over the line y = -5/3x, and finally, reflecting over the y-axis again. That means A3 is:
A3 = A2 * A * A2
So, to find A3, all we need to do is substitute the value of A into the equation and calculate:
A3 = A2 * A * A2
= [[-1, 0], [0, 1]] * (A2 * A2)
= [[-1, 0], [0, 1]] * [[-1, 0], [0, 1]]
= [[1, 0], [0, 1]]
Therefore, the matrix A that induces T, the reflection over the line y = -5/3x, is:
A = A2 * A3 * A2
= [[-1, 0], [0, 1]] * [[1, 0], [0, 1]] * [[-1, 0], [0, 1]]
= [[-1, 0], [0, 1]]
That's it! The matrix A is [[-1, 0], [0, 1]]. Just remember, reflecting over lines may seem serious, but you can always add a little humor to make the math more enjoyable!
To find the matrix A that induces the reflection transformation T over the line y = -5/3x, we can follow these steps:
Step 1: Determine how the transformation T affects the standard basis vectors.
The standard basis vectors in ℝ2 are e1 = [1, 0] and e2 = [0, 1].
Let's apply the transformation T to these vectors:
T(e1) = reflection of e1 over y = -5/3x
Since e1 is a vector pointing in the positive x-direction (x = 1, y = 0), reflecting it over the line y = -5/3x will give a vector pointing in the negative y-direction (x = 0, y = -1).
Therefore, T(e1) = [-1, 0].
T(e2) = reflection of e2 over y = -5/3x
Similarly, e2 is a vector pointing in the positive y-direction (x = 0, y = 1). Reflecting it over the line y = -5/3x will give a vector pointing in the negative x-direction (x = -1, y = 0).
Therefore, T(e2) = [0, -1].
Step 2: Use the images of the standard basis vectors to construct the matrix A.
The matrix A that induces the transformation T can be constructed by arranging the images of the standard basis vectors, T(e1) and T(e2), as columns:
A = [ T(e1), T(e2) ] = [ -1, 0 ; 0, -1 ]
Therefore, the matrix A that induces the reflection transformation T over the line y = -5/3x is:
A = [ -1, 0 ; 0, -1 ]
To find the matrix that induces the reflection transformation T over the line y = -5/3x, we can follow these steps:
Step 1: Determine the images of the standard basis vectors.
The standard basis vectors for ℝ2 are the vectors e1 = [1, 0] and e2 = [0, 1].
To find the image of e1 under the transformation T, we reflect it over the line y = -5/3x. The reflection of [1, 0] over this line can be obtained by finding the perpendicular vector to the line and finding the projection of [1, 0] onto that perpendicular vector.
The slope of the line y = -5/3x is -5/3. The perpendicular vector has a slope that is the negative reciprocal of -5/3, which is 3/5.
To find the perpendicular vector, we can take the vector [3,5] (since it has the same slope) and rotate it 90 degrees to obtain [-5, 3].
Now, we need to project the vector [1, 0] onto the perpendicular vector [-5, 3]. The projection formula is given by projv(u) = (u⋅v / ||v||^2) * v, where u is the vector being projected and v is the vector onto which u is projected.
Calculating the projection of [1, 0] onto [-5, 3]:
proj[-5, 3]([1, 0]) = ([1, 0]⋅[-5, 3] / ||[-5, 3]||^2) * [-5, 3]
= (-5 / (25 + 9)) * [-5, 3]
= (-5 / 34) * [-5, 3]
= [(25/34), -(15/34)]
Therefore, the image of e1 under T is [25/34, -15/34].
Similarly, to find the image of e2 under T, we reflect it over the line y = -5/3x. The reflection of [0, 1] over this line is [0, -1].
Therefore, the image of e2 under T is [0, -1].
Step 2: Construct the matrix A.
To find the matrix A that induces the transformation T, we can form a matrix using the images of the standard basis vectors as columns.
A = [[25/34, 0], [-15/34, -1]]
So, the matrix A that induces the reflection transformation T over the line y = -5/3x is [[25/34, 0], [-15/34, -1]].