# Anisotropic scaling in geometric/Clifford algebra

Take the geometric algebra over $\Bbb R^n$. Suppose we have a blade multivector in this algebra. Now we want to anisotropically scale this multivector.

Is there a general closed-form expression for performing this operation?

I have found similar things for other operations (rotation and etc), but not anisotropic scaling.

Anisotropic scaling is a linear transformation on $\Bbb R^n$ and thus has a well defined outermorphism on $\mathcal G^n$. So if $S$ is your scaling function and $A = a_1 \wedge a_2 \wedge \cdots \wedge a_k$ is your $k$-blade, then $S(A) = S(a_1)\wedge S(a_2)\wedge \cdots \wedge S(a_k)$.
The transformation of a single vector is fairly simple. Scaling by a factor of $k$ in the direction of vector $n$ is done by
$$S(a) = ka_\parallel + a_\perp = \big(k(a\cdot n) + (a\wedge n)\big)n^{-1}$$
The transformation $S(A)$ of a multivector $A$ is the outermorphism of this. (See the other answer.)