Linear algebra: please solve last 2 parts of question 3.3 correctly and handwritten 4.1 ExercisesPS3.1 (MATLAB) Consider the following linear transformations:• T: R² → R² is the linear transformation that rotates the plane 7/3 radians counterclockwise aboutthe origin• S: R²R2 is the linear transformation that reflects the plane about the line y = −x• U : R² → R² is the linear transformation that stretches the plane horizontally by a factor of 3(a) Find the standard motriv 4 for T24(a) B=(b) C =2[14 576 9 12 15111 13624(c) D=(d) E=[12 -3 012 1 136 -5 141][1 1 112 07613 -1 10132114 -2 20 23015 3 35 0 12616-4 56 55 25214PS3.3 (NO MATLAB) Recall that a square n x n matrix is invertible if and only if it is row equivalent to theidentity matrix In.(a) Using language from Section 3.1, list three facts that must be true about the columns of an invertiblematrix A.(b) If T: R² → R² is the linear transformation that rotates the plane counter-clockwise 7/2 radians aboutthe origin, the standard matrix A for T is invertible. Give a geometric description for how A-¹ wouldtransform the plane. That is, explain in words what A-1 should do "visually" to the plane.

PS3.3 (NO MATLAB) Recall that a square nxn matrix is invertible if and only if it is row equivalent to the identity matrix In. (a) Using language from Section 3.1, list three facts that must be true about the columns of an invertible matrix A. (b) If T: R² R2 is the linear transformation that rotates the plane counter-clockwise /2 radians about the origin, the standard matrix A for T is invertible. Give a geometric description for how A-1 would transform the plane. That is, explain in words what A-1 should do "visually" to the plane.

PS3.3 (NO MATLAB) Recall that a square nxn matrix is invertible if and only if it is row equivalent to the identity matrix In. (a) Using language from Section 3.1, list three facts that must be true about the columns of an invertible matrix A. (b) If T: R² R2 is the linear transformation that rotates the plane counter-clockwise /2 radians about the origin, the standard matrix A for T is invertible. Give a geometric description for how A-1 would transform the plane. That is, explain in words what A-1 should do "visually" to the plane.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.1: Introduction To Linear Transformations

Problem 46E: For the linear transformation from Exercise 45, let =45 and find the preimage of v=(1,1). 45. Let T...

See similar textbooks

Similar questions

A translation in R2 is a function of the form T(x,y)=(xh,yk), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x,y)=(x2,y+1), determine the images of (0,0,),(2,1), and (5,4). (c) Show that a translation in R2 has no fixed points.
For the linear transformation from Exercise 45, let =45 and find the preimage of v=(1,1). 45. Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.
Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.
Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
A linear transformation T : R2 → R2first reflects points through the x1-axis andthen reflects points through the line x2 = −x1. Show that T can also be describedas a linear transformation that rotates points about the origin. What is the angleof that rotation?
Find the linear fractional transformation that maps (1,i,-1) to (-1,i,1).
Q.2. a. Find the transformation R (x) which reflect each vector x= through the line along the vector u = b. Show that the transformation R (1) is linear and isometry. c. Draw the vectors x and R (I) forx:
Find the Jacobian J of the transformation
Find the bilinear transformation whose fixed points are 1,i and maps 0 to -1.
Let T/6 : R² → R? counterclockwise by T/6 radians. Let Tref : R² → R- be the linear transformation which denote the linear transformation which rotates a vector in R- across the line xj = x2. x2 reflects a vector x = (a) Sketch a cartoon illustrating what these linear transformations do to the vector ej = (b) Find the standard matrices of T-/6 and Tref- (c) Find a nonzero vector v E R² such that T/6 o Tref (v) = Tref o Tr/6 (v) or explain why no such vector exists.
For a linear transformation w= (atib)z, the point A= (1+j) in the z-plane was transformed to A' = (-5+14j) in the w-plane. Calculate (atjb), the magnification, and the rotation of the transformation.
Define the linear transformation T: R -R" by T(v) = Av. Find the dimensions of R and Rm. A = 1 -2-2 4 2 2 dimension of R dimension of RM Need Help? Read It