3. Let R1,R2 be rotatio 4 i, and let F be a reflection around some axis...
(4) (a) Determine the standard matrix A for the rotation r of R 3 around the z-axis through the angle π/3 counterclockwise. Hint: Use the matrix for the rotation around the origin in R 2 for the xy-plane. (b) Consider the rotation s of R 3 around the line spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a basis of R 3 for which the matrix [s]B,B is equal to A from (a). (c) Give...
Consider a linear transformation F : R2→R2 In lectures it is shown that the reflection in a subspace can be calcu- lated by Rw(u) = 2 prw(u) – u. Use this formula to find the standard matrix of the linear transformation described above. and hence deter- mine the image of the reflection of the y-axis in the line y = 2x.
(e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto B. (0,1) (1;1) (-1,1) (0,1) v=1 1/2 y-axis y=x2 v-axis v -u b ets t) ide ods.a notteog (0,0) X-axis u-axis (0,0) (e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto...
(c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S onto S' (0,1) v-axis V=1 (2,1) (1,1) y =(x-1)2 у-ахis u 1 v=u-1 u-axis (1,0) (0,0) х-аxis (1,0) (c) Let f : IR2 -R2 be given by f(x,)= (a1)2-y1, (-12) Let S, S' be the subsets of R2 as indicated in the picture below. Prove that f maps S...
Let T: R2 R2 be a reflection in the line y = -x. Find the image of each vector. (-3,5) (b) (7.-1) (c) (-a, 0) (d) (0, b) (e) (e, -d) (f) (9)
Three masses of mass M=1.2kg are initially located at positions: r1→=4.0m i+2.8m j −4.4mk r2→=5.4m i+1.0m j^+2.3m k r3→=−2.6m i^−1.9m j^+2.0m k The masses are held in place relative to each other by rigid, massless, rods which connect them to the z-axis. They begin to rotate around the z-axis with constant angular speed. If the system has angular momentum L⃗ =3597 what is the angular speed of the rotation? Answer in radians per second.
linear algebra Let T: R2 R2 be a reflection in the line y = -x. Find the image of each vector. (a) (-3,9) (b) (5, -1) (c) (a,0) (d) (o, b) (e) ( ed) (f) (9)
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х — у). (i) Prove that f is linear as a function from R2 to R2. (ii) Calculatee the matrix of f. (iii) Prove that f is a one-to-one function whose range is R2. Deduce that f has an inverse function and calculate it. (iv) If C is the square in R2 given by C = [0,1] x [0, 1], find the set f(C), illustrating...
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 → R2 be the reflection operator with respect to the x-axis, defined by 21 21 Compute the adjoint operator Lt. Is L self-adjoint? We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 →...
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...