Is Cross Product of two vectors a linear transformation? (Linear Algebra)

Question

• 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:

1. $$T(v1+v2)=T(v1) + T(v2)$$ for any vectors v1 and v2 in V, and

2. $$T(av) = a T(v)$$ for any scalar alpha a.

I also know how to calculate the cross product between two vectors.

• Question:

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?

Answer 1

Believe it or not, the cross product is linear! Let $T(x) = a \times x$ for fixed $a$. Now, I'll show both conditions at once. Choose $x, y \in \mathbb{R}^3$. Now:

\begin{align*} T(kx + y) &= a \times (kx + y) \\ &= a \times (kx) + a \times y \\ &= k(a \times x) + a \times y \\ &= kT(x) + T(y) \end{align*}

Done! So this map is linear!

Answer 2

Even if you consider $$T(p,q) = p\times q,$$ this function is a 2-linear map, i.e when you fix one argument, the function is linear wrt to the other argument.

Therefore, the cross product is more than just a linear transformation, but it is a 2-linear transformation.