How to determine which of the following transformations are linear transformations?Determine which of the following...

Ikunupe6v

Ikunupe6v

Answered

2021-12-20

How to determine which of the following transformations are linear transformations? 
Determine which of the following transformations are linear transformations 
A. The transformation T1 defined by T1(x1,x2,x3)=(x1,0,x3) 
B. The transformation T2 defined by T2(x1,x2)=(2x13x2,x1+4,5x2)
C. The transformation T3 defined by T3(x1,x2,x3)=(x1,x2,x3) 
D. The transformation T4 defined by T4(x1,x2,x3)=(1,x2,x3) 
E. The transformation T5 defined by T5(x1,x2)=(4x12x2,3|x2|)

Answer & Explanation

Anzante2m

Anzante2m

Expert

2021-12-21Added 34 answers

To test whether T is a linear transformation, you must verify that. for some vectors a and b and some constant c 
T(a+b)=T(a)+T(b) 
T(ca)=cT(a) 
T(0)=0 
Example: 
A. T(x1,x2,x3)=(x1,0.x3) 
T(x1+y1,x2+y2,x3+y3)=(x1+y1,0(x2+y2),x3+y3)=T(x1,0,x3)+T(y1,0,y2) 
T(cx1,cx2,cx3)=T(cx1,(c)0,cx3)=cT(x1,0,x3) 
T(0,0,0)=0 
B. T(x1,x2)=(2x13x2,x1+4,5x2) 
T(x1+y1,x2+y2)=(2(x1+y1)3(x2+y2),(x1+y1)+4,5(x2+y2))=(2x1+2y13x23y2,x1+y1+4,5x2+5y2) 
 

Pademagk71

Pademagk71

Expert

2021-12-22Added 34 answers

For T to be a linear transformation, two criteria need to be satisfied:
1. T(x+y)=T(x)+T(y)
T(ax)=aT(x) for a a scalar/constant.
As an example, suppose x=(x1,x2,x3) and y=(y1,y2,y3). Lets
nick1337

nick1337

Expert

2021-12-28Added 573 answers

T(ax)=T(ax1,ax2,ax3)=(ax1,0,ax3)=a(x1,0,x3)=aT(x)

T(x+y)=T(x1+y1,x2+y2,x3+y3)=(x1+y1,0,x3+y3)=(x1,0,x3)+(y1,0,y3)=T(x)+T(y)

Therefore the first transformation is linear as you correctly guessed. Just repeat the same procedure for B-E and see if it works or not.

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