Vector to Tensor properties

[flintbox]

Homework Statement

Suppose A and B are vectors. Show that the object Q with nine components Q_ij=A_iB_j is a tensor of rank 2.

Homework Equations

A tensor transforms under rotations (R) as a vector:
T_ij'=R_inR_jmT_nm

The Attempt at a Solution

I wanted to just create the matrix, but I don't know how to prove that this is also a tensor.

 

[Dick]

Homework Statement

Suppose A and B are vectors. Show that the object Q with nine components Q_ij=A_iB_j is a tensor of rank 2.

Homework Equations

A tensor transforms under rotations (R) as a vector:
T_ij'=R_inR_jmT_nm

The Attempt at a Solution

I wanted to just create the matrix, but I don't know how to prove that this is also a tensor.

Well, how do the vectors $A$ and $B$ transform under rotations?

 

[flintbox]

Thanks a lot!
I think I understand it now:
$$A_i' B_j' = R_{in}A_n R_{jm}A_m$$
$$A_i' B_j' = R_{in} R_{jm} (A_n A_m)$$
$$A_i' B_j' = R_{in} R_{jm} Q_{nm}$$
$$A_i' B_j' = Q'_{nm}$$
So we for proving something is a tensor, we just apply some transformations to it, right?

 

[Dick]

Thanks a lot!
I think I understand it now:
$$A_i' B_j' = R_{in}A_n R_{jm}A_m$$
$$A_i' B_j' = R_{in} R_{jm} (A_n A_m)$$
$$A_i' B_j' = R_{in} R_{jm} Q_{nm}$$
$$A_i' B_j' = Q'_{nm}$$
So we for proving something is a tensor, we just apply some transformations to it, right?

Yes, something is a tensor if it transforms like a tensor.