Is this a Linear transformation

[sara_87]

how do i determine whether the following is a linear transformation:

T1(x,y)=(1,y)

i know that it must satisfy the conditions:
(a) T(v+w)=T(v)+T(w)
(b) T(cv)=cT(v), where c is a real constant
and v and w are real vectors in 2D.
v=(v1,v2) and w=(w1,w2)
but I'm still confused.

Thank you

 

[Ben Niehoff]

So, using the definitions of vector addition and scalar multiplication, plug (v1, v2) and (w1, w2) into your conditions (a) and (b), and see what happens.

(A general hint for all math stuff: Try some things and see what happens.)

 

[mathwonk]

it should take zero to zero for one thing.

 

[HallsofIvy]

how do i determine whether the following is a linear transformation:

T1(x,y)=(1,y)

i know that it must satisfy the conditions:
(a) T(v+w)=T(v)+T(w)
(b) T(cv)=cT(v), where c is a real constant
and v and w are real vectors in 2D.
v=(v1,v2) and w=(w1,w2)
but I'm still confused.

Thank you

Excellent! You just haven't DONE anything with that!

For example, suppose you have v= $(x_1, y_1)$ and v= $(x_2, y_2)$. What is T(v)? What is T(u)? What is T(v+ u)? Is T(v+u)= T(u)+ T(v)?

 

[sara_87]

ok I'm going to try it:
T(v)=(1,v2)
T(w)=(1,w2)
T(v)+T(w)=(1, v2+w2)
T(v+w)=T(v1+w1, v2+w2)=(1, v2+w2)
so the first condition holds.

i'm sure i have done something wrong
?

 

[d_leet]

Why is T(v)+T(w)=(1,v2+w2)?
T(v)+T(w)=(1,v2)+(1,w2)=?

 

[sara_87]

(2,v2+w2)
?

 

[d_leet]

(2,v2+w2)
?

Ok, and is this equal to T(v+w)?

 

[sara_87]

T(v+w)=(v1+w1, v2+w2)
no they're not equal...right?
and how do i show the second condition?

 

[dodo]

Same as before... experiment with a couple of values for the scalar 'c'. You'll notice right away what's going on.

 

[d_leet]

T(v+w)=(v1+w1, v2+w2)
no they're not equal...right?
and how do i show the second condition?

No, by definition of the operator T, T(v+w)=(1,v2+w2). But in either case T(v+w) is not equal to T(v)+T(w). So since it fails one of the conditions for linearity the operator is not linear.

 

[sara_87]

No, by definition of the operator T, T(v+w)=(1,v2+w2). But in either case T(v+w) is not equal to T(v)+T(w). So since it fails one of the conditions for linearity the operator is not linear.

how did you get that T(v+w)=T(1, v2+w2)
could you please show me?


and so if i show one condition then i guess there's no need to show the other..right?

 

[mathwonk]

to repeat myself, since T(0.v) = 0.T(v) = 0, it cannot be linear unless zero goes to zero, but T(0,0) = (1,0), which is a deal breaker.

 

[sara_87]

oh okay, so i guess that's enough to say that it's not linear.

thank you.

i was working on another question on linear transformation and i did it but I'm not sure if i did it correctly:

T(x,y)=(y,x)

again, let v=(v1, v2) and w=(w1,w2)

then, T(v+w)=(v2+w2, v1+w1)

and, T(v)+T(w)=(v2+w2, v1+w1)
so the first condition holds.

AND:

let c be a constant:
T(cv)=T(cv1,cv2)=(cv2, cv1)
and cT(v)=c(v2, v1)=(cv2, cv1)


is that correct?

 

[dodo]

Looks fine to me... but I'd show the intermediate step in your workings of the first condition. Just as you did for the second.

In other words, just saying "for T(v+w) we get xxxx, and for T(v)+T(w) we also get xxxx, so they must be equal" does not show how you got xxxx. Your work for the second condition, on the other hand, shows it well.

 

[sara_87]

i just worked on another one and i think i get the hang of it now but i just want to check this this one to make sure i know how to do it:
T(x,y)=(x,0)

again, let v=(v1, v2) and w=(w1,w2)

then, T(v+w)=(v1+w1, 0)

and, T(v)+T(w)=(v1+w1, 0)
so the first condition holds.

AND:

let c be a constant:
T(cv)=T(cv1,cv2)=(cv1, 0) = cT(v)

is that correct?

 

[sara_87]

I was just doing another example and i want to check this one so i know whether i got the hang of it:

T(x,y)=(x,0)

again, let v=(v1, v2) and w=(w1,w2)

then, T(v+w)=T(v1+w1, v2+w2)=(v1+w1, 0)

and, T(v)+T(w)=(v1+w1, 0)
so the first condition holds.

AND:

let c be a constant:
T(cv)=T(cv1,cv2)=(cv1, 0)=c(v1,0)=cT(v)
so both conditions hold

is that correct?