Question
In: Advanced Math
Is reflection across a line L through the origin an invertible transformation of R 2 ?...
Is reflection across a line L through the origin an invertible transformation of R 2 ? If so, find the matrix representation of the inverse; if not, explain why not.
Solutions
Colby Messinger answered 2 years agoRelated Solutions
Prove that any linear transformation ? : R? → R? maps a line passing through the...
Prove that any linear transformation ? : R? → R? maps a line
passing through the origin to either the zero vector or a line
passing through the origin. Generalize this for planes and
hyperplanes. What are the images of these under linear
transformations?
Prove that any linear transformation ? : R? → R? maps a line passing through the...
Prove that any linear transformation ? : R? → R? maps a line
passing through the origin to either the zero vector or a line
passing through the origin. Generalize this for planes and
hyperplanes. What are the images of these under linear
transformations?
If T : R 3 → R 3 is projection onto a line through the origin,...
If T : R 3 → R 3 is projection onto a line through the origin,
describe geometrically the eigenvalues and eigenvectors of T.
There is a line through the origin that divides the region bounded by the parabola y=8x−7x^2...
There is a line through the origin that divides the region
bounded by the parabola y=8x−7x^2 and the x-axis into two
regions with equal area. What is the slope of that line?
Let T: R^2 -> R^2 be an orthogonal transformation and let A is an element of...
Let T: R^2 -> R^2 be an orthogonal transformation and let A
is an element of R^(2x2) be the standard matrix of T. In a) and b)
below, by rotation we mean "rotation of R^2 by some angle and the
origin". By reflection, we mean "reflection of R^2 over some line
through the origin".
a) Show that T is either a rotation or a reflection.
b) Show that every rotation is a composition of 2 reflections,
and thus that T...
Q2a) Consider a transformation T : R 2×2 → R 2×2 such that T(M) = MT...
Q2a) Consider a transformation T : R
2×2 → R
2×2
such that T(M) = MT
.
This is infact a linear transformation. Based on this, justify if
the following
statements are true or not. (2)
a) T ◦ T is the identity transformation.
b) The kernel of T is the zero matrix.
c) Range T = R
2×2
d) T(M) =-M is impossible.
b) Assume that you are given a matrix A = [aij ] ∈ R
n×n with...
Show that R-1(a) = R(-a). This equation shows that a rotation through a negative angle is equivalent to an inverse transformation.
Show that R-1(a) = R(-a). This equation shows that a rotation through a negative angle is equivalent to an inverse transformation.
9. A line passing through the origin is described by the equation y = Mx, where...
9. A line passing through the origin is described by the
equation y = Mx, where M greater than or equal to 0 is a random
slope. Let (theta) be the angle between this line and the
horizontal axis, in the right side of the plane. Suppose that theta
is uniformly distributed between 0 and 90 degrees (0 and pi/2
radians). What, then, is the pdf of M? What is the expected value
of M?
The resistance of blood flowing through an artery is R = C L r4 where L...
The resistance of blood flowing through an artery is
R = C
L
r4
where L and r are the length and radius of the
artery and C is a positive constant. Both L and
r increase during growth. Suppose
r = 0.1 mm,
L = 1 mm,
and
C = 1.
(a) Suppose the length increases 10 mm for every mm increase in
radius during growth. Use a directional derivative to determine the
rate at which the resistance of...
Let L be the line parametrically by~r(t) = [1 + 2t,4 +t,2 + 3t] and M...
Let L be the line parametrically by~r(t) = [1 + 2t,4 +t,2 + 3t]
and M be the line through the points P= (−5,2,−3) and
Q=(1,2,−6).
a) The lines L and M intersect; find the point of
intersection.
b) How many planes contain both lines?
c) Give a parametric equation for a plane Π that contains both
lines
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