Linear Transformation Homework: Determine Whether Maps are Linear

[ak123456]

Homework Statement

Determine whether the following maps are linear transformations
a) L: R^2 -- R
(x1)
(x2)
--
x1^2 +x2^2
b) L: Mn*n(R)--Mn*n(R)
A-- A-A^T
c)L:P3--P2 f-- f'+(f(3))t^2

Homework Equations

The Attempt at a Solution

I have to show L(x+y)=L(x)+L(y) and cL(X)=L(cx)
for a) i find that (x1+y1)^2+(x2+y2)^2 not equals to (x1+x2)^2+(y1+y2)^2
so it isn't a linear transformaton
for b) can i use the counterexample
(01)
(10)
because A=A^T
for c) no idea for this one

 

[yyat]

b) L(A)=0, but that does not mean that L is not linear.

c) You know how elements of P3 look like: at^3+bt^2+ct+d. Check if linearity holds: L(v+w)=L(v)+L(w), L(rv)=rL(v) where you take for v and w elements of P3.

 

[ak123456]

b) L(A)=0, but that does not mean that L is not linear.

c) You know how elements of P3 look like: at^3+bt^2+ct+d. Check if linearity holds: L(v+w)=L(v)+L(w), L(rv)=rL(v) where you take for v and w elements of P3.

so how can i get a counterexample for b) ?

 

[yyat]

so how can i get a counterexample for b) ?

Try proving linearity instead.

 

[ak123456]

Try proving linearity instead.

thx . i will try it