Let L be a straight line through the origin and denote by Show that: the angle...
4. Let l be the line through the origin which has a directed) angle of inclination 8 from positive x-axis. Let l' be the line through h + Oi with (directed) angle of inclination from positive c-axis. (a) Use re = ROTOR_, to derive the complex equation for the reflection re(2). (b) Use part (a) and re = T re(2) reo T-to derive the complex equation of the reflection
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
Let l denote the line through the origin with direction vector (1,1,1). Let r(t) = (t +1,4, 2t) be a parametrized curve. Compute the point r(to) on the curve which is closest to l, and state the distance from r(to) to l.
Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The projection of R3 onto L — denoted PL : R3 −→ R3 — is definded to be equal to the projection pu onto the vector u. You may assume that PL is a linear transformation. Find the standard matrix [PL] for PL.
In R2 let R be the rotation about the origin through the angle 27/14. Then the matrix [R] representing Ris [R] = Consequently R transforms the point (1, 2) into Check
21. Show that if l is any line through the origin in R3 and x is any vector with its initial point at the origin, then the reflection of x through the line & (acting as a mirror) is equal to 2(proj,x) - x, where r is any nonzero vector parallel to the line (see Figure 1.21). Reflection of x FIGURE 1.21
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
5. (3) Let X = Serie x = ()m - (*) indo 3 x 3 mais Find a 3 x 3 matrix A such that the projection PAX = e vector X ER projected onto the line I that is parallel to a and passes through the origin. Pex - Ax 6. (4) The line l in Ris given by the equation x + 3y = 0. (a) What is the angle e between the positive x-axis and the line...
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
27. Rotations. Show that there is no line in the real plane R2 through the origin which is invariant unde the transformation whose matrix is cos sin sin 0 A(0) COs integral multiple of T. Give a geometric interpretation of this problem commentin when 0 is not an on the case when 0 km for some k E Z. 27. Rotations. Show that there is no line in the real plane R2 through the origin which is invariant unde the...