Determine if F is a linear transformación f(x,y,z)=(x,x+y+z) ?

2 Answers
Jul 9, 2018

Answer:

Please see the explanation below

Explanation:

The tranformation is

#f(x,y,z)=(x,x+y+z)#

That is #RR^3# to #RR^2#

Then ,

Calculate,

#f(x,y,z)+af(x',y',z')=(x,x+y+z)+a(x', x'+y'+z')#

where #a inRR#

#=(x,x+y+z)+(ax', ax'+ay'+az')#

#=(x+ax', x+y+z+ax'+ay'+az')#

#=(x+ax', x+ax'+y+ay'+z+az')#

#=f(x+ax', y+ay',z+az')#

As,

#f(x+ax', y+ay',z+az')=f(x,y,z)+af(x',y',z')#

The transformation is linear

Jul 9, 2018

This can be expressed as a matrix, here #M#, transformation so it is linear:

#F(bbx) = ((1,0,0),(1,1,1) )bbx = M bbx#

Formal tests:

  • Scaling

#bb(F (lambda bbx) )= Mlambdabbx= lambdaMbbx bb(= lambdaF (bbx))#

  • Additivity

#bb(F (bbx + bbx^') )= M(bbx+ bbx^') = Mbbx + Mbbx^' bb(= F(bbx) + F(bbx^'))#