- Background Information:
I am studying linear algebra. For this question, I understand the definition of a vector in $$R^3 => v =(x,y,z)$$, and I know that A linear transformation between two vector spaces V and W is a map
$$T: V->W $$such that the following hold:
$$T(v1+v2)=T(v1) + T(v2)$$ for any vectors v1 and v2 in V, and
$$T(av) = a T(v)$$ for any scalar alpha a.
I also know how to calculate the cross product between two vectors.
Let a be a fixed vector in R3. Does T(x) = a × x define a linear transformation?
- My thoughts:
I don't understand how to show T(a + x) = T(x) + T(a) and T(ax) = aT(x) for a cross product. The fact that numbers are not given makes it confusing as well. How can I approach this problem and prove the cross product is a transformation?