Question

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

Answer 1

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

Answer 2

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^'))`