Question

(a) Find the matrix of the reflection of R^2 across the line y = (1 / 3)x followed by the reflection of R^2 across the line y = (1/2) x. What type of transformation of the plane is this composition?

b) Find the principal axes y1 and y2 diagonalizing the quadratic form q = (x^2)1 + (8)x1x2 + (x^2)2

Answer #1

Find the matrix of the reflection of R2 across the line y =x/3
followed by the reflection of R2 across the line y = x/2 What type
of transformation of the plane is this composition?
thank you.

* Consider the transformations T1=‘reflection across the x-axis’
and T2=‘reflection across the line y = x’. (a) Find the matrices A1
and A2 corresponding to T1 and T2, respectively. (b) Show that (A1)
2 = I, and give a geometrical interpretation of this. (c) Use
matrix multiplication to find the geometric effect of T1 followed
by T2, showing all your reasoning. (d) The product T (θ)T (φ) of
any two reflections T (θ) and T (φ) with angles θ and...

. In this question we will investigate a linear transformation F
: R 2 → R 2 which is defined by reflection in the line y = 2x. We
will find a standard matrix for this transformation by utilising
compositions of simpler linear transformations. Let Hx be the
linear transformation which reflects in the x axis, let Hy be
reflection in the y axis and let Rθ be (anticlockwise) rotation
through an angle of θ. (a) Explain why F =...

Find a 3x3 matrix that performs the 2D transformation in
homogeneous coordinates:
a) Reflects across the y-axis and then translates up 4 and right
3.
b) Dilate so that every point is twice as far from the point
(-2,-1) in both the x direction and the y direction.

Find the matrix of the linear transformation which reflects
every vector across the y-axis and then rotates every vector
through the angle π/3.

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

Let F(x,y,z) = ztan-1(y^2) i + (z^3)ln(x^2 + 8) j + z k. Find
the flux of F across the part of the paraboloid x2 + y2 + z = 20
that lies above the plane z = 4 and is oriented upward.

Create the homogeneous coordinate equivalent of a matrix that
projects a point in R^2 onto the y=x line. Then use your new
transformation on the point (-3,4).

Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?

Find the point(s) of intersection, if any, of the line
x-2/1 = y+1/-2 = z+3/-5 and the plane 3x + 19y - 7a - 8 =0

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