YAMT: How to tell if fcn is a linear transformation?

[Syringer]

I did this a while ago, but can't remember how it's done

i.e. T(x1, x2) = (1+x1, x2)

 

[Muzzan]

If these properties hold for all vectors x, y and scalars k:

T(x + y) = T(x) + T(y)
T(kx) = kT(x)

Then it is linear. Seems like your T fails the first condition (giving a counterexample shouldn't be hard ).

 

[Syringer]

So how would one apply that to the example above?

 

[Muzzan]

Suppose x = (1, 2) and y = (2, 1). We have that:

T( (1, 2) + (2, 1) ) = T( (3, 3) ) = (3 + 1, 3) = (4, 3)

But:

T( (1, 2) ) = (2, 2)

and:

T( (2, 1) ) = (3, 1)

Which gives T( (1, 2) ) + T( (2, 1) ) = (2, 2) + (3, 1) = (5, 3).

Hence T( (1, 2) + (2, 1) ) != T( (1, 2) ) + T( (1, 2) ), so it's not linear.

It fails the second condition too

 

[Syringer]

Ahhhhh, I get it now. Thanks