• 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?

  • 1
    Hint: what is $a \times ( x+y)$? Use the properties! – Sean Roberson Aug 11 at 3:54
  • @SeanRoberson, thanks for the hint, It will be (a X x ) + (a X y), so what would the value of the cross product represent? Also, how can we do this without numbers? – Kourosh Aug 11 at 4:02
  • 1
    First, I recommend changing your scalar from "a" to $\alpha$ since you also define "a" to be a fixed vector. You can see that $T(\alpha x) = a\times \alpha x$ and that $\alpha T(x) = \alpha (a\times x)$. If you can show that $ a\times \alpha x = \alpha (a\times x)$ you are done with that part. Now fill in that blank and write a suitable expression for $T(v_1+v_2)$, $T(v_1)$, and $T(v_2)$ then proceed again. – JessicaK Aug 11 at 4:15
  • @JessicaK, thanks for your help, so having T(αx)=a×αx => αT(x)=α(a×x), and then a×αx=α(a×x) will satisfy the homogeneity property? Or should I add more to it? – Kourosh Aug 11 at 5:58
up vote 5 down vote accepted

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!

  • Oh wow, it is this easy. So the additivity property is done. Would it be enough to show "T(αx)=a×αx => αT(x)=α(a×x), and then a×αx=α(a×x)" for the homogeneity property? – Kourosh Aug 11 at 5:53
  • Notice I also showed homogeneity. "...I"ll show both conditions at once." – Sean Roberson Aug 11 at 6:21
  • My bad, I missed that, but you are the real MVP, I appreciate your help :) – Kourosh Aug 11 at 6:35
  • This proof is kind-of tautologic. The algebraic properties that you use are, in effect, equivalent to the linearity of the cross product. – Federico Poloni Aug 11 at 8:45

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.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.