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#3. Let L:R → Ro be the transformation defined by 1-211 - 312 3:12 - 203...
Find L (4 3 ) if the linear transformation L is the composition of the following...
Find L (4 3 ) if the linear transformation L is the composition
of the following linear transformations: a dilation, with a
dilation factor c = 3, followed by rotation in counterclockwise
direction for an angle α = π, (α = 180 ), followed by a reflection
around the x1–axis.
#1. Find 2(())) if the linear #4. Find L if the linear transformation L is the composition of the following linear transforma- tions: a dilation, with a dilation factor c=3,...
(1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through an angle of 90 in the clockwise direction C. Projection onto the y-axis D. Reflection in...
(1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through an angle of 90 in the clockwise direction C. Projection onto the y-axis D. Reflection in the y-axis E. Rotation through an angle of 90° in the counterclockwise direction -1 0 0.5 0 0 0.5 0 -1 F. Reflection in the r-axis 0 -1
(1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined o...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the...
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the vector obtained by rotating x counter clockwise by 7/3 around 0. Without computing any matrices, what would you expect det (T) to be? (Does T make areas larger or smaller?) Now check your answer by using the fact that the matrix for counter clockwise rotation by is cos(0) - sin(0)] A A= sin(0) cos(0) (b) Same question as (a), only this time let T...
Problem 13. Let l be the line in R' spanned by the vector u = 3...
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...
(1 point) Let f: R3 R3 be the linear transformation defined by f(3) = [ 2...
(1 point) Let f: R3 R3 be the linear transformation defined by f(3) = [ 2 1 1-4 -2 -57 -5 -4 7. 0 -2 Let B C = = {(2,1, -1),(-2,-2,1),(-1, -2, 1)}, {(-1,1,1),(1, -2, -1),(-1,3, 2)}, be two different bases for R. Find the matrix (fls for f relative to the basis B in the domain and C in the codomain. [] =
1. 2. 3. 4. 5. Given that B = {[1 7 3], [ – 2 –7...
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Given that B = {[1 7 3], [ – 2 –7 – 3), [6 23 10]} is a basis of R' and C = {[1 0 0], [-4 1 -2], [-2 1 - 1]} is another basis for R! find the transition matrix that converts coordinates with respect to base B to coordinates with respect to base C. Preview Find a single matrix for the transformation that is equivalent to doing the following four transformations...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system ...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system to be an "object" along with a specific set of modifications that can be performed (dynamically) upon this object. In this case, the object is a bi-infinite straight road with a lamp post at every street corner and a marked lamp (the position of the lamplighter). There are two possible types of modifications: the lamplighter can walk any distance in either direction from...
Find L (4 3 ) if the linear transformation L is the composition
of the following linear transformations: a dilation, with a
dilation factor c = 3, followed by rotation in counterclockwise
direction for an angle α = π, (α = 180 ), followed by a reflection
around the x1–axis.
#1. Find 2(())) if the linear #4. Find L if the linear transformation L is the composition of the following linear transforma- tions: a dilation, with a dilation factor c=3,...
(1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through an angle of 90 in the clockwise direction C. Projection onto the y-axis D. Reflection in the y-axis E. Rotation through an angle of 90° in the counterclockwise direction -1 0 0.5 0 0 0.5 0 -1 F. Reflection in the r-axis 0 -1
(1 point) Match each linear transformation with its matrix. A. Contraction by a factor of2 B. Rotation through...
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the vector obtained by rotating x counter clockwise by 7/3 around 0. Without computing any matrices, what would you expect det (T) to be? (Does T make areas larger or smaller?) Now check your answer by using the fact that the matrix for counter clockwise rotation by is cos(0) - sin(0)] A A= sin(0) cos(0) (b) Same question as (a), only this time let T...
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...
(1 point) Let f: R3 R3 be the linear transformation defined by f(3) = [ 2 1 1-4 -2 -57 -5 -4 7. 0 -2 Let B C = = {(2,1, -1),(-2,-2,1),(-1, -2, 1)}, {(-1,1,1),(1, -2, -1),(-1,3, 2)}, be two different bases for R. Find the matrix (fls for f relative to the basis B in the domain and C in the codomain. [] =
1.
2.
3.
4.
5.
Given that B = {[1 7 3], [ – 2 –7 – 3), [6 23 10]} is a basis of R' and C = {[1 0 0], [-4 1 -2], [-2 1 - 1]} is another basis for R! find the transition matrix that converts coordinates with respect to base B to coordinates with respect to base C. Preview Find a single matrix for the transformation that is equivalent to doing the following four transformations...
1 L, as a dynamical system (Notes from Assignment #2) We take our definition of dynamical system to be an "object" along with a specific set of modifications that can be performed (dynamically) upon this object. In this case, the object is a bi-infinite straight road with a lamp post at every street corner and a marked lamp (the position of the lamplighter). There are two possible types of modifications: the lamplighter can walk any distance in either direction from...
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