MHB Theman123's question at Yahoo Answers (Rotation and reflection)

  • Thread starter Thread starter Fernando Revilla
  • Start date Start date
  • Tags Tags
    Reflection
AI Thread Summary
The discussion centers on two linear transformations in ℝ²: S, which rotates points clockwise through 60° and then reflects them through the origin, and T, which reflects points through the origin before rotating them. The standard matrix for the clockwise rotation is provided, along with the reflection matrix through the origin. It is noted that the matrix for transformation S is calculated as [S]_{B_c} = BA = -A, while for transformation T, it is [T]_{B_c} = AB = -A. Both transformations result in the same matrix, indicating they are equivalent despite the order of operations.
Fernando Revilla
Gold Member
MHB
Messages
631
Reaction score
0
Here is the question:

Let S:ℝ2→ℝ2 be the linear transformation that first rotates points clockwise through 60∘ and then reflects points through the origin.
The standard matrix of S is

Let T:ℝ2→ℝ2 be the linear transformation that first reflects points through the origin and then rotates points clockwise through 60∘.

Here is a link to the question:

Let S:

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello theman123,

According to a well-known property, the matrix of the linear transformation that rotates points clockwise through $60^0$ with respect to the canonical basis $B_c$ of $\mathbb{R}^2$ is $$A=\begin{bmatrix}{\cos (-60^{0})}&{-\sin (-60^{0})}\\{\sin (-60^{0})}&{\;\;\cos (-60^{0})}\end{bmatrix}=\begin{bmatrix}{\;\; 1/2}&{\sqrt{3}/2}\\{-\sqrt{3}/2}&{1/2}\end{bmatrix}$$ and the matrix of the reflection through the origin with respect to $B_c$ is: $$B=\begin{bmatrix}{\cos (180^{0})}&{-\sin (180^{0})}\\{\sin (180^{0})}&{\;\;\cos (180^{0})}\end{bmatrix}=\begin{bmatrix}{-1}&{\;\;0}\\{\;\;0}&{-1}\end{bmatrix}=-I$$ So, $_{B_c}=BA=-A$ and $[T]_{B_c}=AB=-A$.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top