Let e=(a,b,c) be a unit vector in R3 and let T be the linear transformation on R3 of rotation by 180∘ about e. Find the matrix for T with respect to the standard basis e1=(1,0,0), e2=(0,1,0) and e3=(0,0,1). The rotation matrix in R3 by 180∘ is : [−1000−10001] So rotating e by 180∘ gives : [−a−bc] After that how to get the transformation matrix w.r.t the standard basis?

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Let e=(a,b,c) be a unit vector in R3 and let T be the linear transformation on R3 of rotation by 180∘ about e. Find the matrix for T with respect to the standard basis e1=(1,0,0), e2=(0,1,0) and e3=(0,0,1).   The rotation matrix in R3 by 180∘ is :  [−1000−10001]  So rotating e by 180∘ gives :  [−a−bc]  After that how to get the transformation matrix w.r.t the standard basis?

Mazzuranavf · Accepted Answer

Step 1 The matrix that you gave is for a 180∘ rotation about e3. A general way to proceed from there would be via a change-of-basis operation, which involves finding an appropriate basis and performing a couple of matrix multiplications and perhaps an inversion, to boot. Observe, however, that a 180∘ rotation about a vector e is equivalent to a reflection in the span of e, so you can save yourself quite a bit of work by using the formula for reflection of a vector v in a subspace W. This is Rv=2πWv−v, where πWv is the projection of v onto W. In this problem W is the span of e, so for this transformation we have Tv=2eeTeTev−v and we know that ‖e‖=eTe=1, so the matrix that you’re looking for is 2eeT−I3. You can easily verify for yourself that Te=e and that if v is orthogonal to e, then Tv=−v, which is exactly what we want for this rotation.

utgyrnr0 · Accepted Answer

The matrix you give, let&#039;s call it M, is rotation by 180∘ about the vector e3. If you can find an orthogonal matrix T that maps e to e3, then T−1 MT is the rotation by 180∘ about the vector e.

Let e=(a,b,c) be a unit vector in \mathbb{R}^{3} and let T be the linear transformation

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