Extend usual properties of 1D random variables to multidimensional random variables

Question

It is well-known that for two linearly independent, mean 0 $\mathbb{R}$-valued random variables $X$ and $Y$ that $$\text{Var}(aX+bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$$ for any $a,b \in \mathbb{R}$, where $\text{Var}(X) := \int_{\mathbb R} x^2 f(x) dx$ if the law of $X$ (with mean 0) is $f$. I am not sure whether a multidimensional version of the analogous statement if still true. Specifically, if two linearly independent, mean $\bf{0}$ (here $\bf{0}$ denotes the dimensional zero vector) $\mathbb{R}^d$-valued random variables $X$ and $Y$, do we still have something like $$\text{Var}(aX+bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$$ for any $a,b \in \mathbb{R}$, but now with the "variance" defined via $\text{Var}(X) := \int_{\mathbb R^d} |{\bf x}|^2 f({\bf x}) d{\bf x}$ if the law of $X$ (with mean $\bf{0}$) is $f$. I am not sure whether the dimension $d$ plays a role in such generalizations, so I appreciate any help or references.

Answer 1

In the first line you wrote 'linearly independent for '(stochastically) independent'. Linear independnece does not yield any of these equations.

In order to extend this to higher dimensions you need orthogonality :$E \langle X, Y \rangle=0$ This is again implied by indepedence: $E \langle X, Y \rangle=\sum E(X_iY_i) =0$ . Hence we get $E|aX+bY|^{2}=a^{2}E|X|^{2}+b^{2}E|Y|^{2}$ under independence.