Question

determine whether or not the following is a linear map:
U:R^2->R^2 defined by k(x;y)=(x^2;y^2)

Answer 1

so we need to check if\[k(\vec u+\vec v)=k(\vec u)+k(\vec v)\]and if\[k(c\vec u)=ck(\vec u)\]

Answer 2

\[k(\vec u+\vec v)=k(\langle x,y\rangle+\langle a,b\rangle)\]\[=k(x+a,y+b)=((x+a)^2,(y+b)^2)=k(\vec u)+k(\vec v)\]

Answer 3

get the idea @jacobian ?

Answer 4

if the above is true then it has passed a test. if the above is false it is not a linear transformation

Answer 5

it is more like proving subspace right?

Answer 6

sort of, but they are different test. A lot of linear algebra is about verifying certain theorems to check out how some space or transformation or set behaves in some way.

Answer 7

\[k(\vec u+\vec v)=k(\langle x,y\rangle+\langle a,b\rangle)\]\[=k(x+a,y+b)=((x+a)^2,(y+b)^2)\]is that equal to\[k(\vec u)+k(\vec v)\]?

Answer 8

no

Answer 9

so then we are done, though you could check the other theorem for kicks if you wanted.

Answer 10

meaning is not a linear map